For traveling, keep him in as large a container as you can, since you'll be stopping overnight. I like to have my fish in plastic containers in an insulated shopping bag, a lunch box works well for only one fish. Keeps them dark, the temperature is more stable, and there's some padding if they get knocked about a bit. Every hour or so I open the lid of the container to get fresh air in, and I don't feed for a day prior so there's less chance of waste fouling the water. When you stop for the night I'd do a few small water changes, at least one before you go to sleep and one when you leave in the morning. Bettas get shipped across the world all the time in worse conditions, but if you have the space for a gallon of extra water, why not make him as comfortable as possible right.
I personally wouldn't put him on bottled water, is there a specific reason you don't want to use the tap at your school? Fish that are well-acclimated can adapt to some pretty crazy changes, so I'd dig up water quality reports from both districts and see what you're working with. I usually bring a gallon or two of fresh tap water from wherever I'm leaving, so that I can slowly acclimate them to the new water. Bottles usually has had the dissolved minerals that fish need removed, unlike tap, and while you wouldn't need to treat it for chlorine/chloramine, it's also a lot more expensive than tap.
For breaks, it depends on your school's policies. Mine allows students to leave fishtanks for the one-week vacations, but not over winter & summer breaks. So I have to transport my fish four times a year total rather than just once up and once back. A week is not so much time that your fish would starve, and depending on tank size & whether it's cycled it's likely that his water will stay clean enough to last a week. (My cycled 10g was always fine for 1-week breaks) For the longer breaks, I really would recommend taking him with you unless you're staying over for the break yourself. A month or so with no food is pushing starvation limits, but it's the water quality that would kill him. Unless you can invest in an automatic water changing system, it's way too risky in my opinion.