If you want the best for your betta spend the extra money and get a 5 gal with a heater and a filter. A 5 gal is perfect for a betta he has plenty of room to swim around and spread his fins. You could even divide it and keep two if you really wanted.
Put your self in the fishes place. You could live your whole life in your bedroom that was 60 degress but would you want to? No you wouldn't it is the same thing for a betta. He COULD live in a 1.25 gal aquarium but would he like it? No he wouldn't he would like a 5 gal where he can swim around and hide