The Summer is Great, in the Keys!
Most of the out-of-state tourists are gone, road travel is much easier.
There is a mini-lobster (sport)season that hits before the commercial harvesters are allowed,that is crazy fun. And the season for taking them goes on thru the summer.
for the dates. Camping spots are most likely already taken for that week but you can check.
Most of the RV parks in the keys are insanely expensive. You might try:
You can go through www.reserveamerica.com
Here the price is $25 - $30 + tax
The primary site for Florida is: http://www.floridastateparks.org/CampCabinLodge.cfm
Lots of lobster that you can find swimming from shore & an inexpensive short term license can be had at any sports store.
Key West is crazy ... any time of year. If you have never been there, it might be good for a day trip or even on overnight at a small B&B.