In the area your describe, we only know of one place this isn't "dog friendly", that being Cabbage Key and only there because in past, other dog owners didn't clean up after their pets
and/or allowed them to roam freely which irritated other guests on the island and, in some cases, the wild life. In most places beach landings with a pet are best, if possible. For that, at this time of year (considering the cold), carry a pair of sea boots to keep your feet dry and a couple of rags to wipe your dog's feet off once he/she gets, or is put, back aboard the dinghy
and waste bags to collect solid waste for disposal. You might also want to obtain a small folding shovel at somewhere like Bass Outlet that would allow you to bury solid waste in areas where that would be possible such as on Cayo Costa.
One nearby locale you might find enjoyable is Boca Grande. We stayed at the Boca Grande marina on our spring cruise
this year and found it very enjoyable, accommodating; and, pet friendly but you can visit the locale and anchor
in Boca Grande Bayou, just to the north of the entrance channel without charge and there are accessible dinghy docks.
Lastly, if you are not already using it, you might click over to ActiveCaptain.com and register (for free) and use the interactive cruising guide (aka "Live Map") which listed prospective anchorages
with information including "pet friendliness" together with reviews
and comments by other cruisers.