[FoxtrotGolfXray 5.0]

Surprised the most obvious question hasn't been asked: are you insured?

If so, what's your excess? If it's less than the expected repair bill, claim against insurance and let them sort it out. If not, ask yourself how much you're prepared to pay and how much pain and angst you want in dealing with this yourself. If the cost (total cost, not just $, but future impacts on your insurance rate after having made a claim, etc) of claiming it against your insurance is less than the cost to you to handle it yourself, then still claim it against your insurance and let them deal with it.

If you're not insured, or you don't think it's worth the cost of claiming against your insurance, then you're pretty much left to negotiate this out with the other party or just pay up.

At the end of the day, you need to pick whatever is going to be the best avenue for you to resolve it.