In which case cut a piece of perspex to fit your 20 gallon tank in a very snug fit at the sides, drill holes all over it to allow water to flow through, and use this to section off a 10 inh long end of your 20 gallon tank.
Bettas make very poor community fish as you have now discovered.
Keeping one in less than 5 gallons,
espescially unheated and unfiltered is both cruel and unnessecary. These fish may be able to survive in small amounts of water, but this is NOT an excuse for you to keep them that way permanently. I don't care how many years you've been keeping fish, there's no need for it.
If you have space for a 4 gallon jar (which should never ever be used as a home for fish, the surface area is too small and the shape makes heating and filtering a pain in the ###### )
Then you most certainly have space for a 5 gallon tank. It's hardly rocket science to move some stuff around to accomodate a 5 gallon tank. I know for an absolute FACT that a 5 gallon tank will fit in a space just over 12 inches square.
That means it wiill fit on a small table , a kitchen worksurface, a desk, a sideboard. I have one on a small table in the corner of the dining room , it's the hex tank in my signature . The other small tank in the bottom right picture ( the smaller tank on the right ) will fit in a space 12 inches square. I've just measured it so I know I'm right.
I will bet my last fiver you have the space for a 5 gallon tank. Mosty people do. Just shift some stuff.
You also do not need to go out and spend a fortune on a brand new tank. There are PLENTY of places you can get one second hand. Ebay, Garage sales, Car boot sales, craigslist, newspaper classifieds, freecycle. Many of these tanks will be at a very low price, sometimes even free. Many often come with equipment such as heater and filters.
You can MAKE a sponge filter in a few minutes. The link is in this very section. Here is the link
CLICK ME FOR DIY FILTER
You now have absolutely
no excuse not to filter that bettas tank
. None at all.