It's challenging. I have moved leopard geckos over a two day period when I was a kid (like, literally 11-12 when my family was moving lol), but we were able to fit them in their full terrarium into one of the two cars we took and when we stopped at a hotel overnight I could plug in some things like undertank heaters for them that at least tided them over until I could get them to a complete set up at the new place. We sort of took the philosophy that while it wasn't ideal, since it was summer and we were moving through warm states they wouldn't die hideously if they went without for a day or two, and we didn't have to worry about accidentally burning them with things shoved in their tank.
However, even a generous tank for two leos is almost certainly smaller than a tank for one beardie, so fitting that in a car and heating it enough is a tall order. You might consider placing him in a smaller travel tank for the drive (maybe similar to what you'd take him to the vet in?), placing that in a larger cardboard box, and putting hand warmers between the cardboard and the outside of his tank so he couldn't get burned, but would have his tank temperature increased. You would need enough handwarmers to last for your trip, but doable if you can get them.
I also personally didn't feed my leos for a few days before the trip - I was worried they would get stressed or upset by the drive and throw up during it and get even more stressed by that. I also didn't fill water dishes until we were stopped for a bit (say at a gas station) or at the hotel for the night. However, obviously the feeding ecology of leopard geckos and bearded dragons is different so I'd go with your best judgement there.
It would be easier with hotel stops, but I'm not sure what your driving plan is.