Me personally:
Put his container inside a box that is lined on the bottom with bubble wrap(large bubbles if you can find it) it will dampen the vibrations so he is jostled but not vibrating constantly. You can keep it warm with either the warmers put inside say a sock, or cheaper is using a water bottle full of hot water. Every time you take a potty break remove the bottle and refill it from the tap in the restroom. Stuff it down a sock so he can't really get too hot agaisnt the plastic. I'd have 2 bottles so you only have to open his box once to change out the hot one for the cooler one.
Get your rubbermaid type container with bedding and a heat pad(even a person one that you can stick under one end) for if you are stopping. I would not put him in the floor as that is the coolest place normally. Up on a seat secured in place would be better for the amount of vibrations as well. I seat belt my boxes into the passenger seat. A large bowl with a small amount of water would not spill but give him access to water although if he has to go without during the day I do not think it would hurt him.
Myself, a lot of the time I just put them in a bag beside me on the seat or in an open top box, but then I am transporting a short distance. (Actually smaller ones travel tucked in my bra, but we won't discuss that will we? hee hee)
Again, this is only opinion, but I transported a lot of animals including newborn critters this way. Keep the box dark, and the travel container should be rather small to make him feel hidden.
I recently transported my Spider female into town and back and she rode alternatively in my bra or on top of my truck visor. She seemed to really enjoy the truck visor, and I enjoyed the looks from other drivers as she would come out to hang down and check me out once in a while at red lights. Of course she is rather small too.
wolfy(the weird)