I've watched several of my LFS unpacking their betta shipments before. When bettas are shipped to stores or from breeders internationally they are generally placed in plastic bags (and they aren't big plastic bags) which are tied shut, and are then all stacked in styrofoam coolers, which then has a styrofoam lid put on, a heat pack (if it's cold where they are coming from or going) and then that is placed in a cardboard box, taped shut and shipped. They don't get anymore air than what was in the bag at the start of the trip. Those plastic cups you see them in at the store are not what they are transported in, it's what they are dumped into when they come out of the bag.
Admittedly I've done no long drives with my bettas yet, but bringing them home from some stores has been a fairly long endeavor. When I do that they are still in their cups, I arrange them in the passenger seat right up against the part where the back meets, I wrap a towel or blanket (whichever I have handy in my car) around them to help secure them, and then seat belt this 'betta blanket' in. This helps prevent the cups from jostling around, and keeps them fairly dark, which is sid to help keep the betta calmer. How true that is, I don't know. I certainly wouldn't like being bounced and jostled around if I couldn't see where I'm going, but I'm not a betta.
The only fish I've every really taken on any long distance road trips are primarily salt water fish like oyster toad fish, archer fish, sharks, and the assorted other marine fauna like anemones and corals. The fish usually go in five gallon buckets with an aerator stuck in the bucket (bettas don't need aerators). Or in the cases of some of the larger fish (three foot shark does not fit if five gallon bucket) we use 30 and 55 gallon drums to transport them in. A word to the wise if you ever have to transport any large fish around like that, if you're driving in a retired police cruiser with two drums in the back seat - don't get pulled over by cops. If you do get pulled over by cops and they ask you what's in the drums, don't answer sharks! This will save you at least three hours on your trip.
The mason jar in the cooler with padding sounds like a good plan, just secure that cooler to the seat and your betta sounds like it will have a good ride south. I'll be waving at you as drive past me. =]