We spent 6 weeks in the area last year and there are some great campgrounds.
I would recommend the following if you can get in:
A. Riverside RV resort Just west of Port Charlotte. Pet friendly, lots of activities, reasonably priced. Nice size lots.
Riverside RV Resort - A Resort Campground in Arcadia, Florida
B. San Carlos RV resort. Ft Myers Beach, FL. Pet friendly, lots of activities and reasonably priced (smaller lots) with the exception of their superior lots (Big Rig large lots) which do not have a monthly or weekly rate in the winter. Can book for a month but you pay the daily rate.
Fort Myers Beach, RV Park, Camping, Waterfront