But maybe the drain water could be used?
The drain water is going to be 95-100F, so sure it would be plenty for heating a floor to non foot-frying temperatures, but getting the heat transfer without pressure behind it would be an issue, as would the cleaning protocol should it become clogged.
Unless the floor is very large it's unlikely that the floor will ...
"steal" all the heat from the hot water pipe...". Unless you have the largest bathroom in Florida and want to heat the whole floor that's not the issue. The bigger problem would be getting heat into the slab fast enough to make a difference in feel to bare feet. At a typical 2 gpm shower flow and FL type incoming water temps is about 50,000 BTU/hr in winter (less for most of the year) and the conductivity of concrete/gypcrete isn't all that high- it takes quite a bit of PEX tubing to pull even 5000 BTU/hr out of the tubing with 120F water. The thicker the slab the slower it will heat up- it's not clear that it would deliver the floor temperature you're looking for at the end of a 10 minute shower. You'd also be heating the floor in summer when it's 95F outside, which isn't exactly the goal in FL.
Read up on how to design concrete radiant heating slabs- a serpentine of half-inch PEX embedded in a 3" slab with a foot between loops of pipe wouldn't drop the temperature of the water reaching the shower mixer by even 10F, but it would take a pretty long shower to make the floor warm enough to matter.
On the other hand, low voltage electric mesh radiant under the tile can warm the surface temps noticeably over a typical showering time.