In my view, this is one of those situations where you need to do whatever is necessary to protect yourself. If it were me, I would tell the dealer how much I appreciate his offer and let him do the work.
At the same time, I would document everything to MY insurance company, including the offer from the dealer to handle everything. In the event that the dealer does not follow through to my expectations, the opportunity would still be there to turn it over to my insurance company.
The dealer may have the best of intentions, but you never really know till the deal is done.....