How familiar are you with driving in the US? It seems to me that most Europeans underestimate the distances involved from one state to another - you're talking about at least one very full day, or more likely two, of driving. Personally I'd forget about the car and fly from city to city. Flights are generally cheap from Florida to New Orleans, so for the cost of one or two nights' parking at a New Orleans hotel, you could pay for the flight. And as others have mentioned, south Florida has a lot of tolls. I definitely wouldn't drive all the way to IAH (lot of construction and traffic along the way) to catch an international flight.
Looks like BA has some reasonable prices for the nonstops if you are okay with not using Star. I would suggest reversing your itinerary - fly into MSY (the customs will be much faster and easier there) and out of MIA or FLL. BA flies nonstop LHR to MSY on Monday, Wednesday, Thursday, Friday, and Saturday during the summer.