1) Your excellent question was asked by a number of Acharonim. The Sugya of which Shevusim are to be observed, and which ones are permitted in the Beis ha'Mikdash, is a very big one with lots of Machlokos. I will try to bring the principles regarding your question from Haza'ah.

a) We find a few possible reasons why Haza'ah is forbidden as Shevus. Some say that we are concerned that the Kohen will mistakenly carry the holy water four Amos through the Reshus ha'Rabim. Others suggest the reason is that the man who was Tamei, will now be Tahor and it is like "fixing" him and therefore should be forbidden mid'Rabanan like Tikun Kli. Some say that the water of the Parah Adumah is Muktzeh since it has no other use, and it is forbidden to move the water because of the laws of Muktzeh.

b) You mentioned that the Lishkas Palhedrin is Har ha'Bayis, and not the Azarah. This is a great Machlokes between the Ra'avad (Tamid 27a) who holds that the Lishkas Palhedrin was Kadosh like the Azarah. On the other hand, Rashi (Yoma 6a), Tosfos (Yoma 8b), and the Ritva (Yoma 11b) all say that the Lishkas Palhedrin was not part of the Azarah.

c) There is a Machlokes in the Acharonim about whether the principle of Ein Shevus ba'Mikdash applies to Har ha'Bayis as well. This is the Shitah of the Sha'ar ha'Melech (Hilchos Yom ha'Kipurim 1:4). The Mirkeves ha'Mishnah (Hilchos Shabbos 23:4) and others hold that Shevus is forbidden on Har ha'Bayis and is permitted only in the Azarah since the Kohanim are more careful in the Azarah.

d) There is another Machlokes concerning whether it is permitted to do Haza'ah in the Azarah, since the Kohen who is Mazeh the water becomes Tamei, and it is forbidden for a Kohen to be in the Azarah while he is Tamei.

e) There is a Machlokes in the Gemara (Pesachim 47a) concerning whether a Shevus Rechokah -- a Shevus which is not relevant for this actual Shabbos but rather will be used after Shabbos or even the following Shabbos or on Yom Tov which will be during the week -- is permitted to do in the Beis ha'Mikdash. The Halachah is that we do not permit a Shevus which is not relevant for this actual Shabbos.

f) Another principle we find is that we do not permit an act that involves two forbidden Shevusim.

2) After all of this, we can summarize and give a few answers for why Haza'ah is forbidden even if it is a Shevus:

a) If the Lishkas Palhedrin is part of Har ha'Bayis and we hold that Shevus is forbidden in Har ha'Bayis, we can understand why Haza'ah is forbidden in the Lishkas Palhedrin at least.

b) If the Lishkas Palhedrin is part of the Azarah, we can say that it is forbidden to do Haza'ah in the Azarah because if the Kohen Gadol is not actually Tamei and this Haza'ah is only a Chumra (according to Rebbi Akiva), then when the Parah Adumah water is sprinkled on a Tahor person he becomes Tamei until sunset, and the Kohen who does the Haza'ah on the Kohen Gadol becomes Tamei as well, and it is forbidden to become Tamei in the Azarah (Gevuras Ari, Or Same'ach, Cheshek Shlomo).

c) We can say that the Haza'ah is meant for the Kohen Gadol to be Tahor on Yom Kippur which is during the week, so that is Shevus Rechokah which is not permitted (Sha'ar ha'Melech).

d) We can also say that since there are two concerns not to do Haza'ah on Shabbos (carrying four Amos, and Tikun of the Tamei), we cannot permit Haza'ah since it is similar to the principle that we cannot permit two Shevusim in one act (Mitzpeh Eitan, Yoma 8a). Others argue that in this case we are Mazeh the Kohen Gadol just in case, but we have no knowledge that he is really Tamei, and thus the reasoning of Tikun Gavra is not relevant. (Teshuvos Rebbi Akiva Eiger Tanina 114).

Best Regards,

Aharon Steiner