Of course, you didn't ask the GA who okey'd your getting your own hotel to put anything in writing, did you? If you remember the GA's name, you can at least put that in the request, so that she can confirm that she indeed okey'd it.
Where there any independent witnesses? If there were, you may try the small claims court. If not, then there is no proof, only your word, and AA is not gonna give you anything above the amounts in their guidelines.
If there is a high chance you'd fly them again, choose $200 voucher. This way, you're only $20 short. . .
The odds of having a gate agent putting anything in writing are extremely slim, if at all possible. Plus I believe there's language in the CoC that states that it overrides anything an agent might give you.