What adhesive will best take wood stain?

I have a solid-core birch door 48×80 that has some impact damage on one edge, about 3 inches long. There’s an indented area along the edge, and a raised area on one face of the door. I would like to select an adhesive or some sort of filler or something to add to the voided area, then use some oak boards on either side of the door and clamp it to bring in the raised area. After I add the adhesive, clamp it, and let it cure, then I was planning on sanding the repaired area, then staining the entire door. I don’t want the repaired area to look lighter in color because the stain didn’t “take”. I’ve used wood glue before and know that it does not take stain well. Any ideas on what to use would be greatly appreciated. Thanks.
Replies
I seem to recall that polyurethane glue (i.e. Gorilla Glue) will stain up nicely, but I'm not 100% on that. Might wanna wait around for someone with more of a clue to answer. : )
Jason
Is reveneering the damaged stile a possibility?
They make a wood veneer that is adhesive backed, all you do is roll it out with a j-roller then trim to size.
They make it in alot of different species of wood. Might be the best fix.
Agree about veneering if possible.
There are many glues (poly, hide glue, a special one by titebond...) and fillers out there that claim to be staineable. Never worked for me, they never take color like the wood around them.
Agreed... glues never stain consistant to the wood around them.
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