A lot depends upon how you time your departure and arrival, but we've napped at several Rest Areas during the day as well as the far back of
Walmart lots and never had a problem.
I've seen overnighters do it, but I don't feel comfortable after 8pm in public areas.
We've parked at economy motels and rented a room for $60 or so and enjoyed the showers and restaurants. If I
had to stop at Lake City, I'd get a motel and really relax and then tackle the short trip to WDW and be ready to hit the ground running.
Personally, I'm a Road Warrior and if I made it to Lake City, I'd continue onto Paradise. (WDW) We alternate drivers and naps, and often go straight through. (20 hrs) We've stopped a lot at Valdosta, GA and the rates and taxes or lower.
You of course must know your limitations and rest accordingly.
I sleep very little all the time(4 hours per night), but at the first sign of drowsiness or sleepy eyes, I stop driving. As a former cop, I've seen the result of driving while sleepy (DWS).