Be sure to put the snake into a bag (an inside-out pillowcase will work fine) and tie it securely at the top....and lots of padding in the sterelite box with the bagged snake.
I'm sure the warmed seats would be fine. Your biggest concern with this kind of transport is SUDDEN temperature changes...such as bringing the snake out of the warm house to frigid outdoors to a warm car to cold outside to warm school....etc etc. A styrofoam lined cardboard box (or even a little styrofoam beer cooler) might be a better way to carry the snake as it will help keep the temps steady throughout the journey.
I'd just place a handwarmer at the bottom with some padding over the top if it and then the snake on top of that...if its cold enough to need one.