Welcome to the group.
Bettas are one of my favorite fish, I keep a lot of different fish but the Betta is in my top 5 fish. They are a great starting fish and there is so much to learn about them. Many people discount them as think of them as a "Newbie" fish but they are much more. I should get a few more they are just so cool.
Sorry to hear you have medical issues, hope the Betta will help you.
When it comes to moving it will depend on the size of tank you are dealing with. My suggestion would be take the water from your tank and fill a small bucket. Keep the Betta in that bucket with a small air stone(you can get battery powered air pumps) while you are moving. Travel with the fish tank empty should limit the risk of damaging it. After the move refill the tank with the same water and add the Betta back in.
Few more things to keep in mind.
There is a lot of info out there and some isn't accurate. Bettas do much better in warm water (75-80 degrees) so they do need heaters and a filter on the tank.
If your current tank has a heater unplug it about 2 hours before you empty the water and when you refill the tank allow it to sit for 2 hours before plugging the heater back in. That will keep from damaging the heater.
On your filter if you use a simple sponge filter (my favorite filter) you can keep it in the water with the betta and it should be fine, just don't let the filter dry out or it will close it's "cycle".