WHOSE insurance doesn't want to pay for damages?? His? Or Yours?
They don't pay more than the policy limit, in any case!!! You don't buy a Ford and get a Mercedes. He bought $5K of coverage. That's what he gets, regardless of how much the damage is from that accident.
You're going to have to either go through your own insurance company, or sue the other person directly for damages. And just because you win the lawsuit, doesn't mean you'll see any money.
The BEST way to be sure you get paid when your car gets hit, is to buy your own insurance. Want a rental car? Buy rental reimbursement coverage. If YOU don't have collision and rental reimbursement coverage, well, you're underinsured, just like the guy that hit you.
agent, 21+ years