You carrier should not file a claim on your behalf with the other carrier. They cannot (and should not) be acting as your agent in that regard. Also, they would just be getting in way. The two choices are that you file a claim with the other carrier and deal with them direct or you file a claim under your own coverage, have your own carrier address your loss and only then do they submit the charges to the other carrier for recovery.
I guess what I was getting at was what Nyan Cat said. File the claim on your insurance. From there your insurance would then deal with the other company to be re-imbursed. -- Cry "Havoc!" and let slip the dogs of war