Perhaps he takes the name Thu after taking Minas Tirith and the first half of Beren and Luthien?I think that Sauron could choose to call himself Thû and not expect that his enemies are fooled into believing that he is someone other than "Sauron" (a name that hasn't really been established among the Elves yet). I can't think he considers "Sauron" a real name, like Mairon. It's an epithet. He doesn't like it much (although I agree that he could tolerate it). So if he would choose, he'd obviously call himself Mairon - that's who he is. And in the catch-and-release program, there's no reason for him to go by any other name, except in rare cases when he momentarily is fooling someone - but that's more of a mission operative routine, not a change of identity. And I agree that he mostly would impersonate known people. He could use Thû though, as a way to establish himself as a known power outside of Angband. This would not mean that he takes a different shape or anything. It would be a way to make himself a known enemy of the Elves and thereby someone to fear. And as I implied before, I think it could be a name he uses more than once. It could be his necromancer name.
One reason "Thû" could be a name he uses as some sort of disguise is that he could be a little wary of Ulmo's power. Minas Tirith is in the middle of a river and he can suspect that it could be under the influence of Ulmo (not unreasonable). Could the name be a way to avoid being known to the Vala and thereby make himself vulnerable somehow?