Wood stains vary in permeability, but if you let it dry, you're doomed. Any chemical you could use to remove it, will affect the screen fabric as well. You'll be creating a new blotch in eliminating the old one.
Depending on the light gain of your screen, you could also try some automotive white paints, they might be a bit closer in matching. Would give either a more tolerable solution, or serve as a reminder until you replace the screen.
Guard you HT with your life! (assuming a dragon and moat are not an option)