Here is the "official" response from the MPA Lionfish Manager for Mexico:
"Greetings and thank you for approaching to us to answer your questions if it is authorized the capture of lionfish inside the protected natural area Cozumel Reefs National Park (PNAC) as well who can do it.
First of all , I commented you that the correct term to use is the "capture" of lionfish within the ANP because the fishing for any type of marine species is prohibited , the case of lionfish as an invasive species is a problem to be counteract and control more if is a marine protected area.
In the beginning of the invasion we had approach with dive houses and dive staff (instructors , guides, boat captains and sailors ) to train them on the ecological impacts of this invasive species, why is important diminish their populations and the correctly form of capture, first we provide handed net bags and dry bags to promote his capture (Lad Akins REEF suggestion), more recently it has delivered to almost all PNAC authorized boats to to conduct diving activities in the PNAC with E.L.F. ( erradication lionfish ) these were donated by the city council in order to counteract their populations, however there is NO authorization for tourists and general public to capture within the MPA , the above, dive houses know about this regulation and should practice this activity conducted outside the park, because the inexperienced capture, bad techniques and handling of many tourists cause greater impact on reefs that the same lionfish.
The PADI specialty for capturing lionfish with E.L.F. is not over the rules of PNAC, so no one who possesses it can do it. I just want to clarify only divemasters and dive guides who have been previously trained them can remove lionfish from MPA. Other divers can do it outside MPA.
About if the guides and instructors fed native organisms with lionfish, we told them that for handling and management lionfish, preferably left in the water after sacrifice the organisms because the risk increases when they manipulate them more (e.g. get it on the boat to fillet).
However, we also know that the interest and curiosity of the guides to know which native predators can them that has led to the fishes as trigger fish, mutton snapper, eels, even lobsters. I've heard that even nurse sharks eat lionfish and stories of once a nurse sharks followed a group of divers to the surface bringing with it several lionfish.
I've talked with several guides who believe that offering lionfish to native organisms, they will recognize it as a potential prey, however we have emphasized that the above will happen at some point for reasons of natural adaptation in a medium and long term and not short, plus the way the guides offer the lionfish is not correct as native fish will not be found lionfish in pieces, crushed or killed in their natural habitat."
Now everyone knows the official park position on this subject..and you're welcome!
Adam D