Leia was the Senator in ANH? I thought she was just Bail's aid.
I thought Leia says she is a part of the Imperial Senate on a diplomatic mission to Alderaan?
Need to rewatch it.
"I don't know what you're talking about. I am a member of the Imperial Senate on a diplomatic mission to Alderaan..."
There's also this: -
"Holding her is dangerous. If word of this gets out, it could generate sympathy for the rebellion in the Senate."
Again, senators aren't likely to get riled when an aide gets nicked for treason, but if it looks like one of their own is getting picked off their "enlightened self interest" reflexes will kick in. It's also the reason why Vader had his people destroy the ship, send a fake distress signal and lie about it to the Imperial Senate.
All of which is academic since after writing all that, I noted the stage direction in the shooting script specifically refers to her as a senator.

INT. REBEL BLOCKADE RUNNER - HALLWAY
Princess Leia is led down a low-ceilinged hallway by a squad
of armored stormtroopers. Her hands are bound and she is
brutally shoved when she is unable to keep up with the briskly
marching troops. They stop in a smoky hallway as Darth Vader
emerges from the shadows. The sinister Dark Lord stares hard
at the frail young senator, but she doesn't move.