Pretty much what the title asks. I'm going through the Render Unto Caesar questline, and thus have been ordered to destroy the Brotherhood of Steel. if I do so (having completed their questline and gotten the key to their safehouse), does that affect my standing with Veronica, and if so, how?
According to the Fallout Wikia page on Veronica Santangelo she will leave you permanently when destroying the Brotherhood unless you have "high enough reputation", but it isn't more specific than that (see bottom point):
To complete the Mr. House and Caesar's Legion main quest lines it will be necessary to destroy the Brotherhood of Steel bunker in Hidden Valley, which will usually (but not always) cause Veronica to leave the player permanently, even if she is waiting somewhere else at the time. Any items the player gives to her that she has when she leaves the player will stay in her inventory, and it may not be possible to get them back.
It may be possible to get back the gear in her inventory by re-recruiting her (if that option is present), then immediately accessing her inventory via the companion wheel before she has a chance to speak to the player and leave again. You can get the inventory back by pickpocketing her, or killing her when she is walking away.
If the player destroys the Brotherhood bunker prior to meeting Veronica, then tries to recruit her, she will join the player, but almost immediately leave, saying that the player appears determined to fight the Brotherhood.
If the player has a high enough reputation with the Brotherhood of Steel before destroying the bunker, Veronica will not leave.