Yes, let your insurance pay. They will go after the other party with their staff lawyers to try and recoup what they can. They will probably even go after Progressive. They do this all the time and will take all the measures needed, including wage garnishment and license revocation of the other parties if it comes to that. Your premium will not go up as you were not at fault.
Be sure you have collision coverage on your vehicle though. If you only had liability then your uninsured motorist will only cover bodily injury, but if you had collision then it will also cover property damage.
Good luck, and I hope you get a fair settlement.

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Brian
87 300Dturbo 180K #14 head still running R-12 SOLD 12/2017
02 F350 Powerstroke 180K
05 Chevy Express 1 ton w/Royal Utility box 120K
08 Infiniti FX-35 40K
15 Golf Sportwagen TDI 35K
10 Sprinter 3500 chassis with a Class A Winnebago on it. 56K