First two weeks of December should be great, right before the holiday rush. I've seen availability in the Keys State Parks for that timeframe in years past.
My absolute favorite down there is Long Key state park if you can book a site. Long Key is not far from Jolly Roger.
(Jolly Roger is great if you can't get into Long Key and/or you want full services - amenities like a full hookups, pool, laundry facility, cable TV.)
Every site at Long Key faces the tidal flats so you are right on the Atlantic and can watch birds come and go - very peaceful. Check site length, some are just for tent campers. The state parks have rest rooms with showers, sites with water, electric, picnic table and a fire ring...no sewer but a dump station to use on the way out.
There is an RV park called Boyds near Key West but no parks in walking distance of town...will be tough parking a rental RV the size of an A or C in town on the street. There is some municipal parking but if I recall correctly, it is a covered garage with height limitations. Boyd's could give suggestions.
You might consider renting a car at Marathon's airport for a few days for forays to KW and dive trips.
