As you can imagine, Florida's fishing regulations read like an encyclopedia. Your original question is quite a bit different than your follow up post. State residents are typically allowed more freedoms than non-residents.
Your question asked if you could ship within the state. Since that's a pretty odd scenario, I didn't find anything with a quick search. I sort of assumed you would be shipping these fish out to your home address, which is different than shipping out of state. A quick search found this:
It is unlawful to sell, offer for sale or transport out of the state any freshwater game fish unless specifically permitted by the FWC, except that licensed anglers may transport two days' bag limit of legally harvested game fish.
(I interpret that to mean that an out of state angler can take home two day's bag limit. But I believe the exception doesn't apply if you're transporting to keep or sell)
The daily bag limit on this species is 2. A 3 day non-resident license is $17 (pretty cheap actually). Being that PBass are relatively easy to find as an aquariast and not an expensive fish at all, I believe purchasing raised or net caught fish would be more economical than shipping 4 hooked and stressed fish. And then, still not sure if it's even legal.
I would call FWC and ask them specifically. I'm sure you're aware that fines are usually pretty substantial and it's not worth the risk. (no offense, but it somehow doesn't feel ethical either)
Good luck!