I've used Red Oak Miniwax stain on Osage several times and it won't penetrate the ring side of the bow, only the belly side where the grain is exposed. I kinda like the lighter and darker areas coming together. The red oak stain brings out the natural orange and brown in Osage really well. I just wipe it on and use a dry rag to wipe off the excess and it is what it is. In your case dye may be a better option?
I don't know what kind of paint would look good on a wood bow, I've never had the desire to put paint over wood it just seems wrong to cover it up. Maybe you could post a pic of what ya got right now so we can see what your dealing with?
Pictures speak a 1,000 words.
Here's what the red oak minwax looks like on the belly of an osage bow I just completed. I love how it makes the grain pop out and tones down the bright yellow just a bit and gives it more of an orange brown color more like aged osage looks.
In this pic of the back you can see that it does almost nothing at all to penetrate the ring side.