You should really try to avoid coming across the state on 41. This is one dangerous road. 2 lanes, undivided, with not much shoulder. Lots of semi's slowing traffic. People trying to pass multiple vehicles stacked up behing semi's. Alligator Alley/I-75 is a much better alternative. 4 lanes divided by concrete barrier and a nice wide shoulder. The only problem you might have would be law enforcement. I have bicycled on
interstates (although not Alligator Alley) many times without incident, but this is South Florida, and the cops here are a little uptight. The general rule, as I know it, is that if there is no alternate route, then you can proceed on the interstate. If you decide on Alligator Alley, and you get stopped by police, you might want to tell them that you are going no further south than Ft. Laud. If you tell them that you are going to the keys, then 41 would be an alternate route, and they would probably deny access to Alligator Alley. Another possible route would be to ride through Arcadia(you could camp at Peace River Campground) , work your way southeast on county roads to Moorehaven(Fisheating Creek Campground), then on to South Bay and south on 27. Hwy 27 is 4 lane divided with nice shoulder.
If you are still determined to come across on 41, there are a couple of campgrounds between hwy 29(the road to Everglades City), and Miami. Check with Everglades National Park for the details. Also on this route, there will be plenty of trees, so a hammock will not be a problem. Stealth camping will not be a problem either,as you will be in the middle of nowhere. Also, if you are into art, there is a well known, black and white landscape photographer, Clyde Butcher who lives and has a gallery along 41. Well worth the stop. And if Clyde is there at the time, he might even let you pitch on his property. He is very approachable. He has the only private residence in Big Cypress National Preserve.
Last but not least, are the mosquitos and no-see-ums. I don't think you realize what you are in store for. You will be in the middle of a huge swamp, in June, during the rainy season. I will say a prayer for you. Here's a couple of suggestions from a lifelong florida-boy who has spent a lot of time in the Everglades. Buy something better then 'Off". Look for a high concentration of DEET, 60% or better. Have long pants and long sleeves. Columbia makes great lightweight outdoor clothing. Get one of those net hoods that protect your head and neck. No-see-ums bite your scalp, eyelids, and the inside of your ears, they are worse than mosquitos to me. Mosquito coils are somewhat effective, but you have to use a lot of them. Set up camp at least 2 hours before sun-down, when they are at their worst.
Good Luck