Honestly if you have established rock that you feel good about (not in the middle of dinos or some nonsense) I would toss the sand. It certainly makes up a huge surface area for filtration, but you don't have any fish. Sand is cheap and now is a time to use money to reduce the complexity of all of this. Buy some fresh sand ahead of time, but deal with rinsing etc when you get to the new place.
I'm not sure how complicated your life is (partner, kids, lots of difficult things to move, etc) but for me (complicated) I would probably get the tank broken down two days before the move so that I had a day to clean it up, and still not spend the very last day before the move on the tank. "I need to be screwing with my fishtank while you load the moving truck" is likely to end with... decreased domestic tranquility. Moving is chaos. There will be so many non fish tank related things to do.
Reef tank stuff always takes twice as long as I estimate it will, and moving also takes more time and effort than I expect.
I feel quite strongly that you don't need as complicated system as you're describing for the time in transit. Think of what the fish go through to get from Fiji to here. Even from live aquaria to here. You don't even have fish. Coral consume very little oxygen. 4-6 hours in a 5 gallon bucket is nothing. If you let them get to 85 degrees, you will probably regret it.
You could put the corals in bags in a cooler and that would keep the heat out. same goes for the live rock, which I suggest transporting wet, or partially submerged with saltwater soaked clean detergent free towels over it.