The Mishnah in question1
הוּא הָיָה אוֹמֵר, אֵין מַשְׁקִין לֹא אֶת הַגִּיֹּרֶת וְלֹא אֶת שִׁפְחָה הַמְשֻׁחְרֶרֶת. וַחֲכָמִים אוֹמְרִים, מַשְׁקִין. אָמְרוּ לוֹ, מַעֲשֶׂה בְּכַרְכְּמִית, שִׁפְחָה מְשֻׁחְרֶרֶת שֶׁהָיְתָה בִירוּשָׁלַיִם, וְהִשְׁקוּהָ שְׁמַעְיָה וְאַבְטַלְיוֹן. אָמַר לָהֶם, דֻּגְמָא הִשְׁקוּהָ. וְנִדּוּהוּ, וּמֵת בְּנִדּוּיוֹ, וְסָקְלוּ בֵית דִּין אֶת אֲרוֹנוֹ. אָמַר רַבִּי יְהוּדָה, חַס וְשָׁלוֹם שֶׁעֲקַבְיָא נִתְנַדָּה...
He would say: One may not make a converted woman nor a freed maidservant drink [from the "bitter waters" that were drunk by a woman suspected of adultery as a test]. But the Sages say: [They are] made to drink. They said to him: There was the case of Karkamit, a freed maidservant in Jerusalem, who was made to drink by Shama'ya and Avtalyon! He said to them: They made her drink an "example" [i.e. the "bitter waters" they made her drink were not authentic]. They excommunicated him, and he died in excommunication, and the Court stoned his coffin. Said Rabbi Yehudah: God-forbid [one should say] that Akavya was excommunicated...
Were the Rabbis allowed to not do the proper procedure?
The Rambam (to this Mishnah) explains the segue between these two parts of the Mishnah, and he touches on your question in the process:2
ואומר שגיורת משוחררת אין משקין אותה מי סוטה וכשהביאו ראיה ממה שעשו שמעיה ואבטליון אמר שהם לא השקוה מי סוטה על אמיתתה אבל עשו תחבולה שחשבה שהן מי סוטה והם כדמות מי סוטה ודוגמא הוא הדמיון ועשו זה לפי שהיתה גיורת3 והם גרים וכאילו הם לא סברו להרחיקה להסתפחה בדת ישראל וקשה זה הדבר על החכמים על שלעג עליהם ונדוהו:
And [Akavia ben Mehallalel] said that a convert [and] a freed woman - we do not make her drink the Sotah waters. When they brought a proof from that which Shamayah and Avtalyon did [i.e. the incident with Karkamit], he said that they did not make her drink the Sotah waters over her truth, but they made a ploy that she thought that they were Sotah waters, and they appeared like Sotah waters. They had made an "example," and they did this because she was a convert3 and they were converts, and they didn't want to distance her and cut her off from the Jewish religion. This matter was so hard for the Chachamim [to believe] until they cried out over it and excommunicated him.
So, it seems that according to the Rambam, the Chachamim had the same problem with Akavia's opinion that you had. But we still need to understand Akavia's opinion.4
Perhaps Tosfos (Berachos 19a, s.v. דוגמא) can shed some light on this (this is just the relevant part of Tosfos):2
ולא מחקו התורה עליה אלא ליראה אותה
They only erased the Torah for her to scare her.
According to Tosfos, there is no need for anything to be erased into the waters according to Akavia. The main part is that she should be scared if she actually is guilty.
But, didn't my [some number between 1 and 12]th grade Chumash teacher tell us that it was Hashem's Name erased into the waters that gave them their miraculous properties? Sure he did. That's because the Rambam paskens like the Chachamim (Hil. Sotah 2:6).
So why did they not put her through the proper procedure? Well, according to Akavia's position, as understood by Tosfos, they did. There was nothing wrong with doing that.
I would assume, however, that based on Akavia's argument that they were just going through the motions to not push her away, this is not the ideal way of doing things, even if it is valid.
Was it an oath in vain?
אִם־לֹ֨א שָׁכַ֥ב אִישׁ֙ אֹתָ֔ךְ וְאִם־לֹ֥א שָׂטִ֛ית טֻמְאָ֖ה תַּ֣חַת אִישֵׁ֑ךְ הִנָּקִ֕י מִמֵּ֛י הַמָּרִ֥ים הַֽמְאָרֲרִ֖ים הָאֵֽלֶּה וְאַ֗תְּ כִּ֥י שָׂטִ֛ית תַּ֥חַת אִישֵׁ֖ךְ וְכִ֣י נִטְמֵ֑את וַיִּתֵּ֨ן אִ֥ישׁ בָּךְ֙ אֶת־שְׁכָבְתּ֔וֹ מִֽבַּלְעֲדֵ֖י אִישֵֽׁךְ ... יִתֵּ֨ן יְהוָ֥ה אוֹתָ֛ךְ לְאָלָ֥ה וְלִשְׁבֻעָ֖ה בְּת֣וֹךְ עַמֵּ֑ךְ בְּתֵ֨ת יְהוָ֤ה אֶת־יְרֵכֵךְ֙ נֹפֶ֔לֶת וְאֶת־בִּטְנֵ֖ךְ צָבָֽה וּ֠בָאוּ הַמַּ֨יִם הַמְאָרְרִ֤ים הָאֵ֙לֶּה֙ בְּֽמֵעַ֔יִךְ לַצְבּ֥וֹת בֶּ֖טֶן וְלַנְפִּ֣ל יָרֵ֑ךְ
If no man has lain with you, if you have not gone astray in defilement while married to your husband, be immune to harm from these bitter waters of bitterness. But if you have gone astray while married to your husband and have defiled yourself, if a man other than your husband has had carnal relations with you...may the LORD make you a curse and an imprecation among your people, as the LORD causes your thigh to sag and your belly to distend; may this water that induces the spell enter your body, causing the belly to distend and the thigh to sag.
Is any part of this oath not being fulfilled by not having the proper text dissolved in it? Well, remember, according to Akavia, there's no issue with having the "wrong" text dissolved. But even so, read the oath more closely. If she is innocent, these waters will do nothing to her. If she is guilty, she'll explode from drinking them. Ultimately, the issue revolves around how you translate the "bitter waters." While the Chachamim require a Megillas Sotah to be dissolved in the waters to fulfill that criterion, Akavia doesn't, as we noted above. If you're still not convinced, let me quote an earlier part of the above Tosfos that I skipped:
בערוך פירש דוכמא השקוה מי צבע אבל לא היו מרים ממש
In the Aruch [the phrase "they gave her Dugma to drink"] is explained [as] "they gave her Duchma to drink," [i.e.] colored water, but they were not literally bitter.
Again, we see that according to Akavia, the term "bitter waters" can be taken extremely loosely.
Is this even a "real" oath?
Another avenue that should be explored is if this is even an oath in the first place.
The (previously uncited) conclusion of v. 22 states that the woman responds "Amen, amen" to this oath. The Gemara in Shevuos (29b and 36a) learns from this that if Reuven adjures Shimon to make an oath, and Shimon responds "Amen," it is as if Shimon made the oath himself, and if Shimon violates it, he is punished accordingly. The Rambam (Hil. Shevuos 2:1 et. al.) and the Shulchan Aruch (YD 237:2) both pasken thus. So clearly this is a valid oath.
What kind of Shevuah is this?
Just for completion's sake, let's take a look at Hilchos Shevuah.
I'm going to be nitpicky, because this could make a difference. There are two types of oaths that are often confused with one another. A "vain oath" is one who swears regarding something impossible, or regarding something which is known to be false, or regarding violating a mitzvah, or regarding violating a previous oath. An ordinary oath (an "oath of utterance") is one who swears regarding an event in the future or the past, regarding its occurrence or non-occurrence. This is all discussed in the third chapter of Maseches Shevuos.
Technically speaking, the oath of a Sotah would not be a vain oath, but rather your run-of-the-mill oath regarding something happening or not happening in the future.
Well, regardless of what kind of water this is, would it not be a valid oath? If she's guilty, she'll blow up. If she's innocent, she won't. Remember, according to Akavia, the contents of the water don't matter.
Were they liable for embarrassment?
Again, according to Akavia, this was a perfectly valid way of doing the procedure. Nowhere do we find that embarrassment payments are given for an ordinary Sotah, so why should this be different? Further, she would still be eligible to the bracha given to innocent Sotos.5
But... We're not told in the Mishnah that she was innocent. Maybe she really did explode from the Sotah waters. In that case, this question is a moot point.
Is this Geneivas Da'as?
What is Geneivas Da'as?
Geneivas Da'as (gi•NAY•vas DA•as, also gi•nay•VAT da•AT, etc.), noun.
Doing something for somebody for the sole intent of making them think something about you which is false.
Literally translated Stealing one's knowledge
Examples: "One is prohibited to steal the knowledge of any person, even the knowledge of a non-Jew. How so? One may not sell a non-Jew a Neveilah in the place of Shechted meat, nor hide from a dead animal in the place of a Shechted animal, nor may he beg his friend to eat by him when he knows he won't accept..." -Rambam, Hil. De'os 2:6, from Chullin 94a
So, according to the Rambam and the Gemara, you might be correct in saying that this is Geneivas Da'as, since they are giving her dyed water with a fake document rubbed into it, while she thinks that it's real Sotah water with a proper Megillas Sotah rubbed into it.
However, this is still based on the false premise that this isn't real Sotah water. As we demonstrated above, this is real Sotah water according to Akavia, and so this isn't Geneivas Da'as. Sure, she thinks it's a proper Megillah rubbed into it, but ultimately this is real Sotah water.
Still, you could ask based on their intentions: they didn't want her going off the derech, so they faked her out with not-real-but-technically-real Sotah water. Perhaps that is Geneivas Da'as, as she doesn't know that they didn't have to do this, and they're only doing it to keep her on the derech?
Nevertheless, there is a precedent for such a practice.
Mishnah Demai 6:10:1
גֵּר וְגוֹי שֶׁיָּרְשׁוּ אֶת אֲבִיהֶם גּוֹי, יָכוֹל הוּא לוֹמַר, טֹל אַתָּה עֲבוֹדָה זָרָה וַאֲנִי מָעוֹת, אַתָּה יַיִן וַאֲנִי פֵרוֹת. וְאִם מִשֶּׁבָּאוּ לִרְשׁוּת הַגֵּר, אָסוּר:
A convert and a non-Jew who inherit from their non-Jewish father: he [the convert] can say to him: "You take the idol worship and I the money, you the wine and I the fruit." But if they have come into the possession of the convert, this is forbidden.
Avodah Zarah 64a-b:2
The Gemara is discussing what happens when one receives idolatrous items, and it attempted to bring a proof from the above Mishnah in Demai. The Gemara refutes this challenge:
א"ר פפא ירושת הגר קאמרת שאני ירושת הגר דאקילו בה רבנן גזירה שמא יחזור לקלקולו תניא נמי הכי6 בד"א שירשו אבל נשתתפו אסור
Says Rav Pappa: You say the inheritance of a convert? Inheritance of a convert is different, for the Rabbis were lenient regarding it, as a decree lest he returns to his crookedness [i.e. he ditches Judaism for his inheritance]. We taught a Braisa similarly:6 "When are these words said [that the convert can make this stipulation]? When they inherit, but if they are partners, it is forbidden."
For reference, the same Mishnah, as well as Rav Pappa's reasoning and Braisa, are brought down in Kiddushin 17b as well.7
Clearly, then, the Rabbis decreed that Geneivas Da'as does not apply when it comes to keeping a convert Jewish. If that doesn't sit well with you, well, ask away.
1Translation by Sefaria
3If you're unfamiliar with the procedure, when one buys a non-Jewish slave of either gender, they undergo conversion, and for the duration of their servitude, they have the same status in respect to what mitzvos they have as women, and upon their being freed, they are like converts in that respect. Nevertheless, there are several halachic differences between converts and freed slaves (see the last Mishnah in Horayos for one example). As will be discussed several times in this post, there is good reason why she is referred to as a convert rather than a freed slave, as that is the reason why they went through this procedure.
4It should be noted that other commentaries say that the Chachamim thought Akavia was denigrating Shamayah and Avtalyon's honor, and that's why they excommunicated him. This is also the explanation given in Berachos 19a. But the Rambam thinks it's a good question, anyway.
5See Rashi, Bamidbar 5:28, quoting Sotah 26a that an innocent Sotah is blessed to be fertile, to have children more easily, and that they should be more beautiful than the ones she's had.
6I am unfamiliar with the source of this Braisa. While a parallel teaching to the Mishnah does appear in Tosefta Demai, 6:12, its only addition to this teaching is that Onkelos was stringent upon himself and threw his entire inheritance into the sea. It then switches to bathhouses, a discussion dealt with elsewhere in Avodah Zarah.
7While the statement in Kiddushin is attributed to Rava, note that Rava was Rav Pappa's Rebbe. Clearly, then, when Rav Pappa said his statement, he learned it from his Rebbe.