Indeed, the water is the problem (not the stain itself). You could try and see if alcohol maybe works as well, in extracting the color (do not boil alcohol! highly flammable!) You could also, as suggested, sand the bow after the stain, but before the final finish oil/varnish. If you use 600 or even 800 grit, you will only sand the raised grain, and not the stain itself. You could also try to sand the bow AFTER the first coat of oil/varnish. The finish has then, hopefully, protected the stain, and only the raised grain without stain color will be sanded off.
Another option is to moisten the bow with tapwater, PRIOR to staining. Use a wet towel and moisten the entire surface. Let it dry, and sand with 400 grit sandpaper. Repeat a second time. Then, after the water and sandpaper have removed all the grain that could potentially be raised, you apply the water based stain.
Experiment a bit on some scraps of wood. One wood species probably react differently than another species.