Hi Ken,
Here's my advice: do NOT carry an ASP or other collapsible baton until you receive your "CWP". Legally speaking, once you have you "CWP" it's okay to carry one - but you may receive undue attention from law enforcement for doing so.
I believe the ASP to be a "concealed weapon" by its very nature. You have 13-17 inches of steel pipe hidden away in a handle section only 8-9 inches long (those numbers are based on the ASP 21" and 26" batons respectively). If you spend any amount of time studying case law (court decisions) regarding concealed weapons you learn that much of the focus is whether or not a person would have some reasonable opportunity to discern that a potential attacker is armed AND what the nature of the weapon such attacker is armed with. In the case of the collapsible baton, even openly carried, the true nature of the weapon is "concealed" until the baton is deployed/opened.
There is also a strong bias on the part of many law enforcement officers against private citizens owning/carrying/using what is felt to be "law enforcement only" equipment. Although there is no federal nor Florida law prohibiting a private citizen from owning, carrying (with a "CWP") or using (with justification) a collapsible baton - you may nonetheless face this bias if you are contacted by law enforcement.
From a practical standpoint, you should seek out and attend a basic training/certification class. This is so that if you ever have to actually use the baton you can show that you as a reasonable and prudent law abiding citizen had received training in the safest and most effective use of the baton. Translation: attending a professional training class is like buying an insurance policy that may assist you in the event of arrest or prosecution.