Not sure as far as car rental goes, my guess is you could one of the local rental places meet you pretty much anywhere or take a taxi to the rental place.
So for Stuart, you've got the st lucie river, north or south fork, you should be able to leave it there without getting harassed, however in either area, if you can't find someone to watch it and check on it, you may get picked clean.
The other option is in Jensen Beach, on the north side of the Jensen Beach causeway, again I wouldn't want to leave a boat unattended here for two weeks, but it is protected from the weather
for the most part (open from the south).
If it were me and I could afford it I would throw it in a marina, I'd have a hard time not being able to watch a boat or have someone watch it for me for two weeks. The Anchorage in Stuart is gone, won't be open by the time you get here. Vero Beach may be another option if you want to pay for a mooring
ball for two weeks, I think it's pretty safe there. If you want marina suggestions let me know.