Excellent question! In fact, this question should arise from the reading of that section of the Gemara in Shabbat 21b, which states that an elder in the city of Sidon in fact followed the view of Beit Shammai:
אָמַר רַבָּה בַּר בַּר חָנָה אָמַר רַבִּי יוֹחָנָן: שְׁנֵי זְקֵנִים הָיוּ בְּצַיְדָּן. אֶחָד עָשָׂה כְּבֵית שַׁמַּאי וְאֶחָד עָשָׂה כְּדִבְרֵי בֵּית הִלֵּל. זֶה נוֹתֵן טַעַם לִדְבָרָיו כְּנֶגֶד פָּרֵי הַחַג, וְזֶה נוֹתֵן טַעַם לִדְבָרָיו דְּמַעֲלִין בַּקֹּדֶשׁ וְאֵין מוֹרִידִין.
Rabba bar bar Ḥana said that Rabbi Yoḥanan said: There were two Elders in Sidon, and one of them acted in accordance with the opinion of Beit Shammai, and one of them acted in accordance with the opinion of Beit Hillel. Each provided a reason for his actions: One gave a reason for his actions: The number of lights corresponds to the bulls of the Festival. And one gave a reason for his actions: The number of lights is based on the principle: One elevates to a higher level in matters of sanctity and one does not downgrade. (Davidson Edition translation)
Indeed, Ritva there (d.h. echad) writes:
אחד עשה כדברי ב"ש. ואפי' למ"ד שהעושה כדברי ב"ש לא עשה כלום או חייב מיתה כדאיתא בפרק קמא דברכות (י"א א'), שאני הכא שאין מחלוקתם אלא להדור מצוה בעלמא.
One acted in accordance with the view of Beit Shammai: And even according to the opinion(s) that one who acts in accordance with the view of Beit Shammai has done nothing, or deserves death, as is brought in the first chapter of Berachot (11a), here is different, since their argument was only about Hiddur Mitzvah.
In case you thought it was just Ritva (or that Ritva was not enough), Rabbi Yisrael Meir Kagan Poupko understood this to be the view of Rif, before the Ritva was published. In Beur Halacha to Shulchan Aruch Orach Chaim 671:2 (d.h. veyesh) he writes that since Rif brought this section of the Talmud, he understood that one fulfilled the Mitzvah (presumably even in a Mehadrin Min Hamehadrin fashion). However, as he writes there, since no one else had made this suggestion, he would not rule in this fashion.
However, since then, we have merited to the Ritva's commentary, which was printed in 1967 (long after Rabbi Poupko passed away), and explicitly makes this statement. Thus, it is likely that Rabbi Poupko would rule that one has fulfilled the Mitzvah nowadays.
Note: This view is also explicitly taken by Rabbi Mordechai Bannet Rabbi Yosef Shalom Eliashiv, Daf Al Hadaf (all found in commentaries to this section on Shabbat 21b) and Rabbi Shlomo Zalman Auerbach (Minchat Shlomo Tinyana (#58), and is implied by Rabbi Ya'akov Yehoshua Falk in his P'nei Yehoshua commentary there (d.h. sham peligi bah).
(See also Daf Al Hadaf there, which provides further commentary on the story of the two elders, including some "chassidishe torah", but this seems to be the mainstream view.)
It is hard to know whether an omission of this view by Shulchan Aruch and Rambam is meaningful, as they only discuss the view of Beit Hillel. One could argue that it is, from a well-known acronym of the word Chanukah. Abudarham (Hilchot Chanukah) writes that the word Chanukah is an acronym for:
ח' נרות והלכה כבית הלל
Eight candles, and the Halacha follows Beit Hillel
Although we might have the tendency to look at this simply as a cute saying, see Sheyarei Knesset Hagedolah, Orach Chaim 489:3 who writes strongly to defend such statements, and includes this statement in his list.
This statement is also brought by various Acharonim, such as Knesset Hagedolah, Ateret Zahav and Peri Megadim to various parts of Orach Chaim 670, for some, within the discussion of the view of how many candles to light, and for others, as a nice saying at the end.
If we take this view literally, it seems that the various views Berachot 11a should apply. It is beyond the scope of a Mi Yodeya answer to now research exactly the Halachic ramifications of acting like Beit Shammai, and to what degree we follow the different views in Berachot 11a, but it is possible that according to that view, one would not fulfill the Mitzvah of Chanukah candles. Perhaps (B'ezrat Hashem) I will have time to write an article on this someday...
Edit: some further reading can be found here and here (at the bottom of the page).