Like everyone said. This is scumbaggy.

I've been hit before and I've never used my insurance. In fact the first time I ever got hit and called them, they told me to go with the at fault person's insurance first. And that's worked out for me.

Now if they start stalling and giving you BS, I'd call your insurance for consultation and at their advice use your policy. They will then go after the at fault party's insurance.