Generally any insurance claims dealing with the government are an extra pain in the ass and take FOREVER, so be aware of that. Generally when it comes to third party claims you'll want to go through your carrier to get the damage repaired quickly and completely. Once everything has been completed, your insurance company will go after the liable party for damages, including your deductible. If you don't want to use your insurance company, the at fault party's insurance carrier is just going to tell you to repair everything and send them a total bill once everything is done and paid for.

Good luck.