Filing the claim on your policy means you are going to have to pay your collision
deductible and could possibly impact your rate. Personally, I would not agree to do that. The majority of rear-end collisions are considered to be the fault of the driver in the back, so it should not take them any longer to complete the claims process regardless of which policy is covering the damage. If I were in your position I would stand firm and require that the other driver cover your damage.