Accident at fault, no insurance, pls help?

My girlfriend (driver) hit another car when we were tryingg to make a right turn in Miami. It was our fault. However, the car we were driving was rented from Alamo under my name without any insurance for third party. She does not have the insurance herself before. I tried my metlife insurance and they won't cover because her name is not listed. Metlife gave us a estimate $4600 ( I think this is higher than our expect) which is far way beyond our budget. The adjuster said it will be paid by Alamo first, then Alamo will get reimburse from us. Is there any way that we can avoid paying this from our pocket? Pleas advise.