Ok, to correct the Fragarach example:
If Fragarach goes Answerer on Ea, then Ea's attack plainly does not happen. There is no space-time ripping leftover, it would be as if Ea was never used. Only NPs like Gae Bolg would be immune to that effect, since the curse has already guaranteed the result.
Mechanism speculation ahead:
(Even with the reversed order, since Gae Bolg was already invoked, it will hit Bazette's heart. Rather, the effect of hitting the heart was already written into fate, so even when Bazette attacks first, her fate was still sealed. It's an interesting situation where Gae Bolg wasn't activated (and it was activated) but it still hit. Or maybe it's something like, because the fate of being hit was certain, the cause was forced back into play. Even though Fragarach might have erased the cause, Fragarach did not erase the effect. Usually, erasing the cause is enough to erase the effect, except for Gae Bolg, it is the effect that is absolute, the cause follows that. If you erase the cause, the effect will just necessitate another cause.)
Anyways, Fragarach as Answerer is A rank. If you BP it, I guess it might become something like A+ rank? That doesn't matter anyways, the point of Fragarach is the whole reversal of order thing.
As for Kotomine, mind that Masters announcing themselves to the supervisor does not mean revealing information about their Servant (other than maybe class). Likely, Bazette was too dangerous to keep alive? (Not in the fearing for his life kinda way but more like his plan.)
I mean, it's not like he knew Bazette has Fragarach (or maybe he does), and it's not like he would be particularly worried about Gilgamesh, but all the same Bazette is a competent magus and she's his equivalent in the MA. He was an executor, she's a sealing agent. Likely whatever plans he had would have been opposed by her, so she was an obstacle.
I sincerely doubt, however, that the main reason why he killed her was because she might have been able to defeat Gilgamesh.