I agree with the people who say I wouldn't allow it.
However, I think you're sending the wrong message when you point out that
BM didn't contribute to the cost of the car and talking about the financial issues. None of that is of any meaning to the kid, nor should it be.
If you're providing a car for the child, how you pay for it shouldn't be the child's concern. Similarly, what you allow the child to do with it should have nothing to do with who pays for it.
IMHO, as the CP, you have an obligation to parent the child to the best of your ability when the child is with you. Driving a car is a huge obligation and creates massive liability and risk. Therefore, it is perfectly reasonable for you to control that risk. IMHO, stating that the child can only drive within 10 miles of home town unless there's an adult in the car is not unreasonable for a new driver.
However, the child should be able to earn your respect and trust at some point, so you should have some understanding of what is required before the child IS allowed to drive out of town. Once she has reached that level of trust, it should not matter where she is driving to - visit friends or visit BM.
You could tell her that you do not want her driving anyone in the car without your express permission, but that's going to be hard to enforce.
I wouldn't worry too much about BM suing - that's what you have insurance for. Just make sure you pay your premiums on time and make sure that you have adequate liability coverage (in general, it should be equal to or greater than your net worth, but ask your agent for guidance).