I realize I didnt give any advice for dive operators since I dont normally like to cheerlead on here but I'll throw a couple recommendations for lionfish hunting.
@Sloeber 's advice was spot on. Pretty much everyone you ask will give you their opinion on the "BEST DIVE OPERATOR EVER" in Cozumel, just like every restaurant downtown has the "BEST MEXICAN FOOD" in Cozumel. Not every dive operator wants to deal with lionfish hunting outside of the marine park when the bulk of their divers want to do southern sites inside the marine park.
Aldora Divers is a good option as they have HP120s/HP117s/HP100s. It is nice to have the extra gas if you're hunting deep as people do tend to get excited and use more gas when hunting (myself inluded). They also have enough boats to split by experience level and some of their captains know the northern sites very well. You will want to make sure they have a group that wants to go north. Sometimes this is best arranged ahead of time, start inquiring early so that they know you're interested.
Gato's Reef is another good option. They are newer dive operator and used to work for Aldora. They do not have a park permit to my knowledge so the bulk of their diving is up north which they know very very well.
@ChuckP 's Ambient Diver may also go north and do hunting. I am not sure.
This is hardly an exchaustive list but just a few recommendations based on personal experiences with the boat captains and crew.
Variables are wind, current, depth, knowledge of dive sites and in some cases enough willing participants to go north are some
The vast majority of the lionfish from the pictures above were hunted at approximately 135-140' There was no lack of lionfish there but it was more a time management and gas management problem.
