Can Florida remain independent in a timeline where the USA still exists?
I think it can as long as it is in British hands for quite a while.
So lets assume Britian gets Florida either after the ARW or even better the Napoleonic Wars (it gets it as compensation for lossing something else) and an American defeate in the War of 1812.
Britain the props up a colony of Florida just south of the US, who is not very happy about this but can't do much about it (for now) since it just lost the war of 1812 (maybe we get an independent NE in there as well).
The Creek Wars will still happen / follow. The British garrison in Florida uses this to its advantage allying with the Creeks and inviting them to settle in Florida afterwards (to give Florida a small population boost). Britain also settles Florida with some Carribbean colonists and does very little about the runway slaves from the US settling in Floridas interior. Floridas population would still be low. But by 1830 it might be double of OTL's at the time (that's 60,000 though growth would slow down afterwards).
Tensions between the US and Britain would ultimately boil over again. Not just over Florida, the runaway slaves, etc. but over Oregon or some other border dispute as well. War errupts in the middle of the nulliification crisis. The war prevents the tariffs from being repelled, angering SC who still goes about nullifying them and refusing to fight in the war provoking a mini-civil war within the US as the War with Britain is going on. It is 1812 all over again.
Florida manages to hold off a and invasion by using the swamps and rivers to its advantage. While Britain manages to hold off the Americans at sea. The War itself marks Florida's national awakening.
Yes it is a small country - population wise - but a proud one. It gets a second population boost after the "black wars" in the US (an alt-civil war). And by the 1900s it would be elevated to the status of Dominion, or Kingdom (seeing how TTL's Britain would likely not care about upseting the American's by propping up a Kingdom in its borders).
TTL's Florida would marked by a Seminole, British, and black herritage. But by mordern days it would mostly resseble a Carribean country. Despite early realtions with the US, a good chunk of Florida's economy would be centered on tourism (and retirees) from the US and other norther neighbors.