Twice the idea about water based ink. But I'd recommend to try it on a sample first. Felt is usually made out of wool, which is a natural fiber, though I'm not sure if it would work since you've mentioned a coating. Either print it, let it dry, and don't even worry about curing it - since it won't be washed - or run an iron over it to help it along a bit.
Plus the ink will have a lower profile on the table, which should 1) Have less effect on play, 2) Wear better.
Still, I suppose anything that changes the texture of the covering will affect play immensely. So nope, I won't recommend you to get rid of this coating.
