If it has to be one way or the other, personally, I think that if it’s her car, she should drive.
1. It would be weird for her to give him the car keys and tell him to drive, and since in most of the Yeshivish world, it’s mekubal for women to drive, the issue here (as I see it) is about whether or not it’s okay in the first place for either to give a ride to the other, but I don’t think it matters who the one driving is.
I don’t know if it’s a problem or not for either one to be giving a ride to the other, but if there’s an issue, I would think that would be it.
2. Safety-wise, it makes more sense for the girl to be the one driving.
3. By giving him the keys, she is making the whole thing into more of an issue, and I think that’s worse.
Either way, the non-driver should sit in the back.
If you think it’s really an issue, ask your LOR. This is just my humble uneducated opinion.