Yes I have to agree with everyone without a heater your Betta could get very sick I would not risk it very bad idea. I hope you can figure something out for a your project maybe get a loan for a family or a friend.
Remember the Light at the end of the tunnel may be You.
When I first started with bettas I use to keep them w/o heaters...and believe it or not all of them did fine. If you're under a budget then I would just be cautious not to let the tank fall < 70F. If above 70F then you should be fine.
I wouldn't recommend removing the filter. However, when I lost power and heat for 10 days from the hurricane a few months back I used to heat water up on the stove and fill empty water bottles with it and float them in the tank to warm it up it their normal temp. I would also wrap towels around the tank to help keep the heat in. It was the most annoying thing in the world because you have to keep doing it every few hours, but I guess if your adamant about removing the heater you could try that.
Yeah that might work, but it probably wouldn't be kept at a very steady temp. The point is that though it IS possible for your livestock to survive without a heater, it is not recomended. Especially because you are not in an emergency situation with starting your new project.