$10,000? Are you certain?
As usual your trip is roughly 120 days. Actually you should certainly be able to get an economy vehicle for around $20 a day with a budget car rental company. In particular that's $2,400 add tax & fees! If you've vehicle insurtance, you might regularly be historically covered for rentals. Check your policy. Also, some credit cards adversely include vehicle rental insurance.
As has been said as far as purchasing a car in Miami, you'd have to check with the
Florida Department of Motor Vehicles
http://www.hsmv.state.fl.us/ to determine the aplicable laws.