Eli Chaps has done a few up in birch, I think however aniline dye was recommended to me at one time when I was refinishing the 30TKs birch stock. Because birch has a grain that is unique in properties a sander/sealer coat is recommended, this seals the grain with a clear coat so the stain, if used, will come out more uniform in color and blend in nicer. Check the reference section under woodworking to see if there's anything there to help you. Mr fixit
Did a search for that thread, here are some excerpts.
Just an fyi, but oil based stains on that birch stock will not get very dark. You might try a water based aniline dye that will penetrate and give you some color. A friend of mine had a M1-A that had a birch stock and we used a water based aniline dye and it really looked great. We found a color that I think was called Winchester Red. Those are great little rifles. I had one that had seen quite a bit of use that I traded for a 336 SC. I would love to have one in the condition yours is in.
Regards, Augustus
Fiebings Leather Dye works great.
Ditto what Swampman said, I've heard good things about Fiebings, I just couldn't remember the name when I posted.
Regards, Augustus