First up, I would apply for a FLA non resident CWL...I have one myself.
If you list Fla as your state of residence and you put on the app that you have a NY permit when Fla cross checks with NY you run the risk of having your NY permit revoked for there are no nonresident permits in NY and if Fla is your state of residence then your NY permit is invalid.
I have had a Fla nonresident for years and its a good one to have.
I would apply for a Fla non resident its not a big deal to get one.
My "training" course consisted of a hunter safety certificate issued by NYS Dept of Conservation.......courses are available all over for young hunters....cheap, and is accepted by Fla.