I've been in the Ten Thousand Islands in August, but that's just because being eaten alive by mosquitoes and torn apart or electrocuted by afternoon thunderstorms is preferable to being at work.
Seriously, though - it will be hot in September, but if you are somewhat offshore and moving, mosquitoes might not be intolerable, but bring gallons of repellent anyway.
As Keith said, watch the weather reports a few days ahead of time for hurricanes. And no matter what, there will be afternoon thunderstorms everyday, so have an exit strategy.
Having provided those advisories, the trip you suggest sounds utterly amazing. I would love to do it. If you could get someone to drop you off in Key Largo and have the truck / trailer waiting on you in Key West, that would be ideal.
I would suggest studying charts carefully. It's shallow and reefy close into shore around the keys, so knowing your route would be critical
To avoid that sickening crunching sound.
In terms of staying in hotels along the way - I have always thought it would be really neat to reserve a hotel room on the beach and then pull up in the TI instead of the car.
For camping, once you get down around Islamorada, there are a number of uninhabited islands, but I can't speak for their campability.
Let us know about your plans as you get closer to the date, if for no other reason than so we can live vicariously through you