Okay, so, the obvious - Carol is hiding her family ties to Admiral Marcus. Why? And how and when does Kirk and co. learn otherwise? Is the British accent (I know, I know, it's Alice's real voice) part of the act?
I don't think she's necessarily hiding anything.
I think I read somewhere that Wallace is supposed to be her mothers name. Maybe she and daddy are having a few issues and she's not using his name as a result.
Or she doesn't want people to think she's advancing in Starfleet because of her name and favors from the Admiral.