In British English, I've heard "bird" used to mean a jail term. In this particular context, the following would make sense:
"If I got caught [with the gun] I would have said no comment and took the bird"
That is to say, she wouldn't have said anything about why she was carrying the gun, or on whose behalf: she'd have simply gone to prison and served the mandatory sentence that goes with carrying an unlicensed firearm.
I would have said no comment and took the bird with me
does not make sense. It could be that the journalist (or their editor) didn't understand what the interviewee was saying, and has, during transcription or editing, tried to shoehorn in the "with me" in order to make it parse in their mind. Or, the interviewee said it, either as a particular idiomatic usage, or just having lost track of the sentence and added something incorrect on at the end.
And thanks to Matt Эллен for the prompt that "with me?" is used as a shorter form of "are you with me?", at the end of a sentence, to mean: "do you understand?":
I would have said no comment and took the bird; [are you] with me?