No way will a betta and dwarf puffers survive together long. Thos yummy fins will become a target sooner rather tahn later.
I have found my most succesful bettas to have been in heated tanks with live plants and weekly water changes of at least 50%. Of course I have a heavily planted 5 gallon tank. If the tank were 2 gallons then I would be doing a water change of 50% twice a week (that is only 1 gallon at a time so not too much work for you).
A heater may be hard to get in a smaller tank, so either find the most stable heated spot in the house or maybe look at getting a heated reptile mat or sometimes you can buy heated mats to grow seedlings at garden stores.
Ideally a 5 gallon tank with a heater and small bubble type filter would be best if you have the space and money (though they are pretty cheap).