As an update, Coinpot relented and returned the money. They still thought I was a scammer, but conceded the possibility that it was a mistake. I suppose I have to admit I was wrong about them looking for any excuse to not return the money.

That said, my advice to anyone else stuck in my situation is not to rely on emotional appeals. Be polite, but point out the possibility that they made a mistake, and ask if there is any way they can review other details of your account, or what they would accept as proof of your innocence.