Quick question on water based dyes, like those from Bingham, after cutting my limb down to size I want to dye the wood between the glass to match the glued sides. Would this dye effect the glass in any way or would it just ruboff?

You want to dye the wood/lams prior to glue up. Use water based or alcohol only stains only.

I would dye them prior, but after cutting the limbs down the now exposed wood would not have dye.

No problem with dyeing the edges after cutting the bow down.