Stopping overnight is best for both you and horses.
That many miles will take you better part of 2 days, as you can figure 50 miles an hour average, which will account for traffic, and fuel/bathroom stops.
I stopped 2 times coming 1000 miles, overnighted at 2 different barns and had no problem with horses loading. Since it wasn't a familiar place to them, that helped too.
And I would never haul in a slant untied, as if horse decides to snake head under partition, is that is how yours is set up, if you are going to only have him in there, AND can secure the divider so that it doesn't come loose, that would be all right then.
And please do not do the rest area thing, that is one of the most dangerous places you can stop at, especially at night. Lot lizards and drug dealers frequent them many times, and just a lot of unsavory characters hang around them.
Call and get a feel for some of those horse motels. There are quite a few that are B and B's, so you would be right there with horse.
Horses make me a better person.