If you do your research ahead of itme(I'm sure it won't take long to find a charter online), you may be able to find a charter who would be willing/able to take you out, and then when you get back, pack, freeze, and ship your catch in dry ice. If you have someone at home while you're gone to receive such a package and get it to the freezer, it could still work. I've heard of people doing it from the Key Largo area; I have to assume there are charters in Key West who do it too.
Another option for something to do--I found Key West to be a good place for biking. When I was down there, I would always bring my own bike, so I don't know if there are rental places or things like that. But if that's up you're alley, I'd check into it.
Lastly--consider a trip to Dry Tortugas National Park. Contrary to what many people think, Key West is not the furthest west you can go in the Keys--it's just the furthest west you can go by either car or rail. But there are plenty of islands further west, many of which are part of Dry Tortugas National Park. There are regular excursions by both speedboat and by plane which take people to the park's headquarters, which is in an old fort, Fort Jefferson. Makes for a fun day trip. Touring the fort is very interesting, and the snorkeling from there is top-notch!