I have no experience of dealing with the USPS. In dealing with an insurance company, I was able to avoid using a lawyer simply by sending a Without Prejudice letter to them which said "I'm only claiming for my bike frame and my helmet, and not for the cost of hiring a rental car to get to work in the meantime, and I haven't incurred any legal fees yet, so just write me the check now and we don't have to bother with litigation or similar messing about". I enclosed a bike shop estimate for replacement/repair. They sent a check within a week.

Considering that car accidents can be $2000 for a damaged fender and a tow, they're probably going to be more interested in concentrating on fighting off bogus car claims than your legitimate bike claim.

Top tip: be incredibly nice to the person handling your claim. They deal with nasty/angry people all day and a little bit of honey can expedite your claim/convince them that you're legit and there's nothing for them to gain in slowing things down.