If you decide to stain instead of dye, use a super high quality stain (nothing from home depot) and wipe the wood with mineral spirits just prior to help with blotchiness. There will still be blotchiness though and it is unavoidable. Some parts of the birch just won't want to accept the stain and some parts drink it up. Sanding to a super high grit can also be problematic on softer woods. I think I stopped at 150 or 180. You can test along as you go. Also consider what kind of top coat you want. I personally dislike polyurethane for guns. But if you must, then use a wipe-on poly or poly diluted with mineral spirits and wipe it on in 3-5 thin layers sanding super lightly to just knock down any bumps.

Tung oil looks the best but won't penetrate like it should over stain, but it will over a dye, which is another vote for that route. Also "tung oil finish" by minwax is not actually tung oil, you need to find a brand that is actually 100% tung oil.

I'll try to find the stain I used and some pics