What value was there in making the kohen take an oath? If he was a tziduki then he was a kofer in Torah shbaal peh and even if he did the avodah according to chazal it didn't change his beliefs. Wouldn't the avodah be posul or meaningless since it was a performed by a kofer?
1. Rebbi Akiva Eiger cites a similar question to yours in the name of the Pri Chadash. He asks, "How does the oath help? It could be that the Tziduki would take a false oath in order to perform the Mitzvah of the incense the way they believed it should be done?" The Pri Chadash answers that the oath might still deter the Tziduki because since he knows in advance that he would have to swear, he might take seriously the question of taking a false oath. (See Shavuos 39a, that the whole world trembled when Hash-m commanded not to take His name in vain, and see Tosfos to Bava Metzia 5bm DH d'Chashid, that this is the reason why even someone suspected of stealing is not suspected of swearing falsely.)
2. However, I suspect you are asking a slightly different question -- that even if he should do the actions of the Avodah properly, nevertheless his beliefs should make him unfit, in the same way that a Nochri is invalid? However, it seems that the answer is that the Tzidukim were in fact better than Nochrim. See Rambam, Hilchos Shechitah 4:16, that the reason why the Shechitah of a Nochri is forbidden is that since he does not believe in Torah she'Ba'al Peh, he might do the Shechitah wrongly, but if he slaughtered in front of us correctly then we may eat it. It seems that the fact that he does believe in the written Torah means that he does have the Din of a Yisrael. The Chazon Ish (18, DH v'Ikar proves from the fact that, in the Gemara here, they accepted the oath of the Tziduki, that a Tziduki was not considered a complete Kofer. The Chazon Ish implies that since Tzidukim were particular to keep all og the Mitzvos written explicitly in the Torah, their Avodah in the Beis ha'Mikdash would be acceptable as long as they did it in accordance with all of its laws.
(1) Doesn't the Rambam say elsewhere that the tziddukim in fact didn't really believe in Torah shbichsav either but they couldn't admit that publicly and so just acted as if they beleived?
(2) re schechita one could distinguish that schechita is to remove the din neveila hence it would be OK if done by a supervised nochri. However the Avodah requires proper kavanos for the mitzvah itself not just a pro forma action.
(3) re the Chazon Ish how is it possible for one who is kofer in Torah sh'baal peh to "be particular to keep all the Mitzvos written explicitly in the Torah" without the Torah shbaal peh one can't possibly keep mitzvos written in the Torah (eg they would do melachah on what we know as shavuous)?
1) I am not aware of such a statement of the Rambam. However, the Gemara in Yoma 19b,does say that a Tziduki said to his son, "My son, even though we are Tzidukim, we are afraid of the Perushim." The Gemara in Nidah 33b also says, "Even though they are the wives of the Tzidukim, they are afraid of the Perushim and show their Nidah blood to the Sages." However, the Rambam seems to understand that they did believe in the Written Torah but were afraid of admitting publicly that they did not believe in the Oral Torah. See also the Rambam in Hilchos Avadim 6:6 who writes that the Tzidukim of his time possesed the same status as the Kusim in the time of Chazal, before Chazal proclaimed that the Kusim are equivalent to Nochrim for all matters (because they discovered that the Kusim were worshipping idols on Har Gerizim; see Chulin 6a).
2) The Mishnah in Chulin 13a states that the Shechitah of a Nochri is a Neveilah. Rashi there says that this applies even if he did it according to the Halachah and others watched him. The Rosh in Chulin (1:5) cites the Ri who says that this is derived from Devarim 27:7, "You shall slaughter and eat" -- one is allowed to eat the meat only of someone who is commanded on slaughtering.
3) I think your comment on the Chazon Ish, YD 2:18 DH v'Ikar, is certainly a valid one and this matter requires further research. However, it seems that the Tzidukim did not do Melachah on Shavuos. See Menachos 65a which states that it was the Baisusim who beleived that Shavuos always fell on a Sunday. When the Rambam, in Hilchos Temidim u'Musafim 7:11, discusses the latter sect, he writes that these mistaken people left Klal Yisrael. In contrast, in Hilchos Avodas Yom ha'Kippurim 1:7 he writes that the Tzidukim did not believe in the Oral Torah but he does not say that they left Klal Yisrael, so it seems that the Tzidukim were slightly better than the Baisusim.
Back to trying to understand what the Chazon Ish writes, that if someone is particular to keep what is stated explicitly in the Torah, even though he does not believe in Chazal's interpretations, his Shechitah is Kosher. The question was how is it possible that someone who does not believe in Chazal can keep what it says explicitly in the Torah; for example, he will do Melachah on Shavuos (as per Menachos 65a, that the Baitusim said that Shavuos always falls on Motza'i Shabbos)?
It seems that one must say that since the Baitusim erroneously believed that Shavuos always falls on Sunday, this was their idea of what the Torah states explicitly and they were faithful to that, if only according to their mistaken understanding. They always observed a day of Shavuos even though it was on the wrong day, but this is not considered as contradicting what the Torah states explicitly.
One can make a distinction between a "Mumar," someone who has "changed" his faith, and an "Apikores," someone who still believes in the written Torah but understands it incorrectly. The Shechitah of this kind of Apikores is Kosher according to the Chazon Ish. With this distinction, one can answer the question of the Lechem Mishneh on the Rambam, Hilchos Shechitah 4:14. The Rambam writes that if the Shochet is an Apikores (i.e. he denies the Torah and Moshe Rabeinu), then he is equivalent to an Akum and his Shechitah is invalid. The implication is that it is only if he denies the Torah that his Shechitah is invalid, but if he believes in the written Torah but denies Chazal's commentaries the Shechitah is still valid. The Lechem Mishneh points out that according to the Rambam, the Shechitah of Tzadok and Baitus should be satisfactory, but he asks, how does the Rambam know this latter Din?
According to the Chazon Ish, the question of the Lechem Mishneh can now be answered (see Sefer ha'Mafte'ach, in the Frankel edition of the Rambam, who points this out). The Chazon Ish cites proofs for the Rambam's position:
1. Chulin 3b, which says that the Shechitah of a Kusi is Kosher even though he does not believe in Chazal's words.
2. Our Gemara, which says that the Kohen Gadol could be a Tzeduki as long as he swore that he would do the Avodah on Yom Kippur correctly.