Caleb, yes the liguid Rit dye will work direct from the bottle in a wipe on/wipe off situation. If you are wanting to dip with this dye then de-natured alcohol can be added to create a larger volumn.
Mike, you may have capatibility issues with the dye and paint but red dye/stain will be the worst for bleeding.
Been using Rit dye for staining arrow shafts for over 30 years. Learned long ago that if I wanted a solid color crown from paint that I did not stain the area where the crown was to be applied.
Also, remember when using clear that is wiped on, the wiping action can be the cause of bleeding. And when dipping a clear coat make sure it is thin, almost the consistency of water, so the excess runs off quick enough so no build up is created. Furthermore, 3-4 thin coats is much preferred to 1 or two thick coats.
When I used gasket lacquer would have a total of 6-7 coats of clear on my arrows. When I use polyurethane I'll have 4-5 coats.
Use OOOO steel wool in between each 2 coats and then the final one.