Honestly, I think it's way too much of a stress on the Goldfish and the Betta to keep them together. Yes, Bettas will become very depressed in cold water and it WILL make them sick and/or reduce their lifespan. All of my Bettas either have a heater, or are in a heated room... the only times I've had sick Bettas is when they were in <70* tanks.

I think, if your Mom doesn't like so many tanks, you should buy either a 2.5 gallon or 5 gallon Minibow (OR, a 10g, if you want to be generous) and divide it for the 2 Bettas, and add a small heater/whisper filter. They'll be MUCH happier, and you will be down to 1 tank instead of 2! You could also move one of the Betta tanks into the living room, and decorate it nicely... that way you really won't have as many tanks in your room