Check out electronics surplus stores, such as
Skycraft (physically located in Orlando, FL), as they may sell wire by the foot.
If you click on the link ^, some stuff is sold online, but there's usually a lot more in the store. So it's better to visit in person if you've something like this is local to you.
As it happens, Skycraft has 22/3 twisted PTFE by the foot (
here). Stranding is different (19/34), but I suspect this won't be a problem for you. There are also listings for 20/3 and 20/4 that could be of interest, particularly for colors, should the 22/3 not be to your liking.
Hope this helps.