I made an old white glass maple riser Bear look like this with alcohol dyes....not the best result..but it was better than bright white.
The bow needed a refinish anyway...
Sanded thru the finish.
dyed the glass and wood
sanded a little dye off the glass to get the grey color I wanted (would've been BLACK without sanding)
refinished.
You can run alcohol dyes over some finishes and get a little adhesion. Normally methanol would be the solvent of choice for this. Then recoat with your clear.
...some experimentation would be needed though.