Geico was very pleasant to deal with on the phone, and they agree that as long as their policyholder (the other driver) confirms my description of the accident, they will consider him at fault and will pay for everything I need.
They don't have a local facility, so they're going to have an adjuster come take pictures of my car at work. Normally, they'll make payment arrangements the same day. At worst, I'd have to lay out repair costs and be reimbursed.
So now I'll make arrangements with the dealer to carefully inspect the car to see what damage there is, obvious and otherwise, that needs fixing.