I think a lot of this is luck in what gate and TSA agent you get. I had a similar experience last year (in pre-Katrina MSY not Mexico) and was left with no ID or documentation at all.

Ticket agent issued me the return ticket after I explained (in great New Orleans detail) how I lost my wallet. TSA let me through with the same story...no ID at all. I was actually pretty suprised it was so easy to get through without any ID.

Admittedly it was a return leg of a flight, but it still shouldn't have been quite that easy. So maybe with a bit of luck with the agents you get this won't be too tough, especially if you have some photo ID.