We actually RV (fulltime) with our Betta, Fishwish. I put him in his original cup for the times that we are driving. I check the temperature often with a temp gun....and we usually keep the RV toasty (also travel with a cat, a cockatiel, a bearded dragon and hermit crabs! Ha Ha!).
If we are traveling in cold weather, I sometimes plug my little Hydor flat heater into our inverter (you can get them for cars too, just plug em into the cig lighter) and put it close to his cup. We've been over quite a few bumpy roads and Fishwish has been fine. I swear sometimes he likes the crazy ride. I fed him a couple of blood worms during a bumpy ride once and he had a ball chasing them around with the water sloshing, haha!
At that time the water had spilled a bit which was why it was sloshing. I find the less room the water has to travel, the calmer it is. Fill it to the top and make sure you have airholes. I always open the container once in while, but I'm sure it's not all that important if you have airholes....