A photo copy of your Kentucky CCL will suffice for proof of Firearm Competency.
Appointments at the regional FDACS centers are currently 6 weeks out + about two weeks processing. Mailing in to Tallahassee take about 3 weeks processing.
So I would order the application and fingerprint card now, fill out and send in before the move. Once you get a Florida address, you can do an online address change.