Yes, Flywheel. The problems is - you need to get a handle on that guy. He won't just give you his HO or other liability insurance papers.
Since you know he is covered under AMA's policy - this is probably the easiest way to go.
And they all may even deny the coverage after getting the facts. But once the insurance has paid for the bills, they would have to go to their client to get their money back, not the injured guy.
If they deny the claim before they had paid anything, the injured had to sue the pilot to get his costs covered. And that can take forever and he may get nothing.