As a frequent traveler with guns, especially on a trip that far, I would fly. If nothing else the cost of round trip tickets may be even cheaper that the cost of rental car, gas, and food for the trip, and you get to add a couple days on each side of the vacation that would otherwise be lost to travel.
I remember how it was before I was a LEO trying to travel with a gun and have it as accessible as possible, and it would be much more of a pain now.
If you do drive, check over the vehicle every time you stop. There are some LEO out there who give themselves weird challenges, such as competing with another LEO to be the first to get a "felony tag light", which basically means you go out and stop cars with out tag lights till you get a stop that leads to a felony arrest.
If a traffic stop does occur, don't be standoffish, or try any of the crap you see on YouTube. It actually gives the officer reasonable suspicion to extend the stop, and you have no idea what the guy that had that rental car before you did with it. About half the drug traffickers we catch are in rental cars, to the point that when I rent a car I have one of our dogs run around it before heading out.