Rechargeable batteries in the carry-on and in a "device" designed for them (camera, light, charger, etc) or in an FAA approved container. Put all disposables in checked baggage. For what batteries cost in Cozumel, I think there is a big $ grab going on every time they find any kind of battery. I mean, has anyone ever heard of a disposable alkaline battery catching fire or exploding on its own? Maybe if I held a blowtorch to it for 5 or 10 minutes.
No dive lights in carry on the way out of CZM is a new one on me. I got so sick of the battery BS my new light is an LED Princeton Tech that runs on 4 basic, everyday C batteries. I bring 4 in the light down and 4 in the checked bag. I generally replace about 1/2 way through a 2-week dive trip which is far too soon but they'll all go in the trash before I fly back (good or not) as I don't need the hassle. A couple of years ago I had a set of rechargeable batteries that were in original sealed packaging on my way out of Coz as I never needed them and I caught crap over them so I broke open the original packaging and stuffed them in the open charger (where they were far less secure and prone to fall out) and said, "Now they are in a device" and they let me proceed... The logic in that, well, there is no logic, just a perfect display of blatant stupidity and worthless air regulations.