When I moved from South to North I put my fish in food grade containers (I have been advised of people loosing fish with non-food grade, as the plastic can leech) with clip down lids. I wrapped them in towels to keep in the heat and kept them by my feet in the car for the three+ hours. I was also advised to open the lid occasionally to let in some fresh air. The aquarium was at the time very small so I kept the gravel in but removed all the water. Yes, keep your filter media in water to keep the bacteria alive. All fish survived.
Another way to transport the fish as you are only going a short distance, is ask your LFS for some of their bags, this way it is easier to float them in the tank once it is back up and running again.
Is it possible for you to get your bigger tank you wish you had now for your new home?

Any chance you can get access to your new home in advance to set up the tank for the fish, so when you move them it is just a case of transporting them?
NB:You can get heat packs from walking shops that will re-charge, and generally stay warm for the time needed - I use these when I collect fish from my LFS.
WARNING: I almost set fire to the house when I left the pan of heat packs boiling away and the pan ran dry! Lost three heat packs and a pan! This was when I was on the Forum writing about my sick Betta - so the lesson is - do not become distracted when re-charging heat packs!
Out of interest what fish do you have to move?