I tried felt in the beginning, with the same idea $$ as you. Nothing stuck to it. I think in bible school, they used felt pieces on felt
I have also pieced my working wall flannel and wasn't happy with it at all. Although that was the gridded stuff that some designer was pushing at the time.
Save your pennies (look at all that fabric on your shelves you bought and haven't used!) and get the right tool for the job.
When I was away from home for several years, I pinned batting (warm and natural) to the living room wall and it worked very well.
whatever you use put it on tight-ish as you'll want to roll it with masking tape, they get really thready.