That doesn't sound right to me but if it came from the horse's mouth, I guess you have to believe it. My concern would be that the person you spoke to didn't fully understand your problem. I think some good additional points have been brought up.
At this point I think you should mail your luggage if possible. That alleviates one problem, though its probably expensive. My next step would be to call AeroMexico (or better yet email, so there is a record) to verify that they will indeed let you board. Your issue will be having a connection with a totally different airline that is a day after you land. Even though its a 16 hour layover a big sticking point might be that its date is the next day. Like someone else said, every time I fly international they check my visa twice. I think there is a large fine if an airline flies you to a country and you can't get in due to no passport or visa.
With those 2 things completed, I would just fly there, buy a club pass, maybe sleep in the hotel, and lay low. Since you will be there for 16 hours certainly shifts will change and no personnel will probably realize or even care for that matter how long you have been there. If you are paranoid after 8 hours go change your shirt, shoes, and put on a hat. They will never know.