Need advice - accident on road trip out of state

They all will give you the option of going through them or handling the claim through the other party's insurance company. If Amica determined you're not at fault and informed you if they couldn't recover from the other party's insurance company that it would count against you, you need to find a better insurance company.

They're in a much better position to pursue damages against the other party and have greater leverage in the recovery than you will ever do. There have been too many horror stories regarding the other party's insurance company dragging its feet on even minor claims, including those posted here. Insurance companies don't mess with each other (or recoveries both ways would disappear into thin air), but bad insurance companies may try to drag their feet on individual-pursued claims.

While that's true, you need to follow whatever your own policy dictates if you do file a claim through your own insurance company. The other poster's policy that they need to have the car appraised within 24 hours of the accident in order to qualify for rental simply sucks and is indicative of a bad insurance company. This is what happens when you cheap out and shop solely on prices.