As a veteran of roadside trailer tire changes IL recommend replacing ALL your tires with Trailer tires. They are available in radial versions as well. Make sure they are rated to carry your load (your current ones are deficient in that regard). It is a royal pain to change tires on the road, especially far from home, plus, long stretches of highway driving will build some serious heat in those passenger car radial. These two factors combine to STRONGLY recommend new tires before the trip. If I were you, I would buy 8 and carry a second spare (with or without the rim). Finding a matching tire to replace your spare can be difficult on the road and if you lose one on the way down you will still want a spare for the rest of the trip. Good luck, whatever you decide. Remember, free advice is worth every penny.
Larry