Pianotech

If you get the wood wet with any water-based cleaner, it will swell the surface a little. Finishers often do this with water, let it dry, and then sand it smooth again. That makes the surface less apt to raise the grain when doing the final finishing coats. If you use something like alcohol, it might have the same effect as water because it has water in it. It might make the stain soak into the wood more than it already has. I doubt that the stain will ever be totally removed.
I don't know if raising the grain is going to affect the bridge function when it dries. Adding a finish over the area might affect it as well. Or maybe not. I do know that once I tried using epoxy putty on a bridge, or some kind of plastic, and the tone died. You want the wood in contact with the string to be firm and not rubbery.
Alternatively, you "could" route out the section of the bridge and add a new cap. Or use some touchup color to paint over the dark areas.
I got a lot of information by Googling "Removing Wine Stains from Wood".
Many possibilities..