Cool, thank you very much for explaining how the "deposit" works!
I returned the car on Saturday. I called Amex on Sunday, and they could not dispute the charge yet because it was still pending. I checked my account just now and it has now been posted.
I am a bit worried that somehow if I dispute the charge as Amex suggests, this will antagonize Avis and they will actually be less likely to follow through. But I am very naive in all these matters so I realize this is probably pretty silly and I should probably just do what the Amex rep suggested.