You do know that beech and birch are different trees entirely? On top if that he is working with a European variety of beech. Use the barrel canal. Its the same wood. Completely hidden and already on hand. About as simple as it gets.
Yes KY Jon, most of us know the difference between beech and birch. Well, all except for BrentD, who can't even spell beech... and Clapper Zapper, who advised the OP to do a stain test on an entirely different wood species, and then to attempt some eclectic mix of materials and methods to achieve what I'd expect to be a garish result that might satisfy someone looking for a camo finish.
This is the internet, and we must always be careful to sort the wheat from the chaff.