As pointed out in the question, because of the use of the connecting conjunction (את) in the phrase, "דֵּעָה֙ אֶת־יְהוָ֔ה", it is clear that דעה is a verb. And as Rashi and others point out to the source of Rambam's quote in Isaiah, it is from the concept of דעת. The verb root would then be י-ד-ע.
This is classified as Gizrat Peh-Yud, Lamed-Gronit (with the letter Ayin), which makes this a somewhat rare, double-weak verb structure with a Peh-Yud.
The letter Heh at the end of the verb is only supportive of the object in the phrase and indicates that the object is feminine.
In the context of the complete sentence from Isaiah and also the usage by Rambam in the Mishneh Torah which is discussing the final redemption, the phrase should be future tense, which in terms of the form of the conjugation and nekudot suggests that the Binyan is Pi'el (intensified action), 3rd person, singular, masculine. Because of the unusual double-weak root, the opening letter Yud appears to drop leaving only Dalet-Ayin (דע). It's worth noting that this particular verb form only appears once throughout all of Tanach in the Isaiah quoted.
Considering the complete phrase brought by Rambam, the letter Heh appears to be indicating (meaning somech to) the name of G-d (שם, which can be either masculine or feminine in Hebrew depending upon the context).
It's worth noting that this also relates to the concept of Shem being feminine as used in the first word of the Shema prayer. (שמ-ע, the intention is to draw the 7 middot of Imma (intellect), represented by the large letter Ayin, into Nukvah (physical material expression within creation), the feminine as daughter, which is the concept of Shem, name.)
When considering the actual meaning of the verb root, Machberet Menachem in connection with the two letter root ד-ע, indicates that the root has several meanings, including the idea of uncovering or revealing (גלוי), and also actual union like in sexual intercourse, where husband (G-d is referred to as איש מלחמה in the Song of the Sea) and wife become one flesh.
In addition to these ideas, Sefer HaShorashim of Rabbi Yonah Ibn Yanach quotes Sefer Otot HaRifayon and associates it to Iyov 9:5 and the concept of feeling or sensing tangibly.
And the Sefer HaShorashim of the Radak also brings these ideas and adds in relation to the posuk from Isaiah that it also has a connotation of how He is the source of this action, particularly because of the usage of את in the phrase.
And this leads to trying to understand the concept of name as used here, like the name of G-d.
The concept of name as used in the Torah is not the actual essence of that which it refers to, meaning in this case the essence of G-d (עצמות אין סוף). It is only a vessel, a tool so to speak, intended for others to connect and relate to the one, meaning the essence, associated with that name.
This idea is discussed in chassidut and many other places with the terms עצמות ומהות in regard to the first of the ten commandments (For a discussion of the distinction between atzmut and mahut see the beginning of Sefer Dikduk HaGra regarding the different categories of verbs. One relates to space/makom and the other to time. In context, how G-d transcends both time and space.) which begins with Anochi (אנכי), which is not a name but a reference to G-d at His essence (אנכי stands for אנא נפשית כתבית יהבית, I give this ketubah freely, meaning without compulsion.) which transcends the concept of names altogether. This is sometimes called Ain Sof (אין סוף) in kabbalistic literature. And G-d at His essence identifies Himself to us (in His ketubah, the luchot) via the name used in that first commandment.
In this context, the He, meaning the masculine operator of the phrase (remember the binyan of the verb is 3rd person, masculine, singular), refers to G-d at His essence.
Through G-d manifesting the union with His name, meaning the feminine part of this phrase indicated by the ending letter Heh in דעה, which is associated to G-d's four letter name, there is a revelation or uncovering of G-d's essence, meaning what transcends all of creation, within the world, which will be physically and materially sensed and perceived over all of His creation, like the waters of the sea cover over. This is the meaning of the words which Rambam brings in Hilchot Melachim 12:5.
This is often expressed by the saying of our Sages: For the sake of the unification of the Holy One, blessed be He and His Shechina (Divine Presence). It is also the meaning behind saying: Blessed be He and blessed be His name.
It might seem like an outlandish idea to think that Rambam in his Mishneh Torah, a book of psak halacha, would be communicating such a deep, mystical and profound idea (a view expressed by Rabbi Meir Ibn Gabbai in Sefer Avodat HaKodesh), that doesn't seem to be the plain meaning at all. Until you consider what Rambam writes in his introduction to the Mishnah Torah, an introduction which was actually written after he had completed the entire work.
In section 1 of the introduction, Rambam explains that the entire Torah was given to Moshe Rabbeinu at Har Sinai both the written portion found in Tanach and the entire oral Torah, meaning all explanations including the entire Talmud, every Midrash and all the Kabbalistic and Chassidic teachings (Ma'aseh Merkava and Ma'aseh Bereshit). Nothing was excluded.
And then Rambam goes on to explain in section 41 of his introduction that he has intentionally written this entire oral Torah within his Mishnah Torah, in a very precise, refined and abbreviated language to such an extent that if read and understood correctly, no other book about the oral Torah would be necessary to anyone.
And Rambam goes even further in regard to this idea in section 42 to his introduction to say that his intent in writing the Mishneh Torah was that a person should first read the section of the written Torah and afterward read the corresponding section in the Mishneh Torah and from this would know the entire oral Torah and not need any other book.
This also means that Rambam intended the Mishneh Torah to be read in connection with the daily and weekly portion of the Tanach and that it was composed in parallel to the weekly reading.
And just like the beginning of the written Torah, parshat Bereshit, begins with the details of the beginning of Creation for example, including the concept of Adam knew Chava (אדם ידע חוה), the Mishneh Torah also deals with that same subject in Hilchot Yesodei HaTorah chapters 1 through 4 (for example see 2:9-10 and 4:9-13 which also deal with this subject of Da'at).
And looking at the conclusion of the written Torah, which describes preparing to enter the land of Israel and establishing G-d's Kingdom through the King of the Jewish people. (Note also Devarim 34:10-12 which like the beginning of the Torah, again mentions the subject of Da'at (See Rashi there) and also the Yad HaChazakah, another name for the Mishneh Torah which contains 14 books (י״ד). These are the subject of the conclusion of Rambam's Mishneh Torah also.
That like the saying of our Sages, The end is wedged in the beginning and the beginning is wedged in the end. And this is also alluded to in the last letter of the written Torah (ל) together with the first letter (ב) of the Torah (which is started immediately upon the conclusion of the final parsha) which totals 32, a reference to the 32 times G-d is mentioned in the Creation (like is discussed in the introduction to Sefer Yetzirah).
And this idea expressed by Rambam in his conclusion to the Mishneh Torah, Hilchot Melachim 12:5, that the whole of the Torah, meaning the union of both the written Torah (like the expression of the Zohar 2:90b:3, 3:35b:8, Tikkunei Zohar 65a:1 that He, meaning G-d at His essence, and the written Torah are one, דאורייתא וקדוש ברוך הוא כולא חד) and the entire oral Torah (which is compared to the Shechinah, G-ds presence in all worlds via the mitzvot, also called His causations, איהו וגרמוהי חד, like is found in the Introduction to Tikkunei Zohar 3b, Sefer Eitz Chaim 42:5, Tanya Igeret HaKodesh, end of 26:4 and Kuntress Acharon 4:1), whose literal seal is ל״ב, meaning Heart (which Rambam associates with the King of Israel, Melech HaMoshiach (Hilchot Melachim end of 2:5)), is the ultimate purpose of Moshiach and the entire Torah.
And this is what is expressed by The prophet Yechezkel 36:26 which says:
וְנָתַתִּ֤י לָכֶם֙ לֵ֣ב חָדָ֔שׁ וְר֥וּחַ חֲדָשָׁ֖ה אֶתֵּ֣ן בְּקִרְבְּכֶ֑ם וַהֲסִ֨רֹתִ֜י אֶת־לֵ֤ב הָאֶ֙בֶן֙ מִבְּשַׂרְכֶ֔ם וְנָתַתִּ֥י לָכֶ֖ם לֵ֥ב בָּשָֽׂר׃
The idea behind emphasizing the connection between the very beginning of creation and the days of the Messiah is because although it looks like a complete change in the natural world, something clearly contested by some opinions in the oral Torah (See for example Mishneh Torah, Hilchot Melachim 12:2 אין בין עולם הזה לימות המשיח אלא שעבוד מלכיות בלבד), Rambam is emphasizing the view of the Gaonim that the change is only returning the creation to its original pristine condition. This is like is expressed in Malachi 3:6 which says:
כִּ֛י אֲנִ֥י יְהוָ֖ה לֹ֣א שָׁנִ֑יתִי וְאַתֶּ֥ם בְּנֵֽי־יַעֲקֹ֖ב לֹ֥א כְלִיתֶֽם׃
Because I am G-d, I have not changed, etc.