That's the appendix carry I mentioned might be good with a flat thin Keltec 380. But, you know what, a stainless steel derringer would probably survive the sweat better. Less moving parts, too. No slide.FamilyFirst, if you have your FL CWL, and you want to remain legal, a good start for answers is in FL Statutes 790.06 Section (12)(a).
I'm unable to find the specific link to the "mental health services" caveat restriction for hospitals, though. I just remember it as one of those extra rules...
Regardless of legality, for the practical portion of your question: get a dedicated belt and AIWB holster for your P365. Just put the belt on over your undies, and wear the scrub bottoms over it as appropriate. A sturdy nylon gun belt holding only a holstered handgun is surprisingly comfortable when wearing stuff like scrubs, workout shorts, swimming trunks, whatever.
Google "p365 aiwb" and multiple options will pop up for your consideration.
Wear it at your 12:30, and the only person who will know it's there is your wife.