I don't believe there is any access to the St Johns South of Jax.
I'm not aware of any tax issues with an out of State boat with a valid and up to date registration from their home State.
If you're looking for protected dockage (hurricane) then maybe think the Okeechobee waterway. There are several good spots/dockage/yards along the waterway that are at least as if not more protected than many places on the St Johns. Being in the middle of the State you can cruise either Coast, and the Atlantic outlet is in Stuart, 250 miles South of Jax and much closer to good jump off spots to the Bahamas and 87 miles direct to West End.
Best place to jump off depends on where you're going. Miami-Bimini is the closest at 50 miles, Abaco's by way of West End on Grand Bahama from Palm Beach or Ft Lauderdale is 70 miles +-. Exuma's are further South but you have to cross and clear somewhere.
You can jump off from anywhere in Florida, these are the closest with Customs Offices.