Your heater should keep the water temp fairly stable even in the cooler room.
I heat my house with a fireplace and while the tanks are in the same room-they are across the room about 50feet from the heat source and its pretty cold over on the aquarium side...maybe 50-60 degrees and my tanks-even the 1gals will stay in the 76-77F range. I use the 78F preset type 25w (Tetra brand) on my 1-10gal unfiltered soil based heavy planted tanks and have not had any issues. On my 20g-75gal NPT's with the adjustable heaters set at 80F will stay in the 77-78F range. I use 2-100w in the 55gal and 2-200w in the 75gal-On the 20 and 25gal's I use 1-75-100w.
Water temp stay pretty close to what I want, however, in the lower level-especially under plants it can sometimes be as much as 10 degrees cooler-especially in the deeper tanks...a bit less in the smaller tanks.
All my Bettas seem to be fine and stay healthy as well as eggs/fry don't seem to have any issues either.
I have found with healthy Bettas the gradual temp change related to room temp, day/night, lights on/off and at different levels to be tolerated.
What watt heater do you have-you might add a second heater since it is a 20gal long-as well as a second thermometer at either end.