Canadian here. Rabbit is about the worst pelt to use in a hat. It doesn't shed water, and it doesn't breathe well. I know because I bought and wore one when I couldn't afford better. Beaver is expensive, but it is the best. In the middle, muskrat or fox are good. They will do a better job than rabbit, but beaver is tops.

There are all sorts of options to fur, including thinsulate and wool. Is it cold enough to wear a fur hat where you are?

There's a company called Acton that makes these for the civvy market. The surplus might be cheaper, but anyhow, this is what I wore flying and snowmobiling in Labrador during extreme cold weather:

https://www.herooutdoors.com/canadian-armed-forces-arctic-mukluks/