First, why would I by a gun for someone whose background I am not confident about? 

Second, if I am unaware of their prohibited status it is not illegal for me to thus -unknowingly- transfer a firearm to them per Federal nor State law. Mens rea is required, it isn't a strict liability offense.

Third, if I'm buying my theoretical felon girlfriend a gun, I would probably at least be aware if she wasn't able to go on vacation with me due to being on the no-fly list. What on earth kind of hypothetical is that?