Seriously Jerry... that is so great you're doing that. BTW that is walnut.
I don't know what the time constraints are for this but.....
You may not want to hear this.... but I would really consider re-doing that dark part.
It's really not that hard. Use some extra fine(the finest grit available where you get it) steel wool and just rub it until it lightens up to the shade you like better. There is no need to do any more staining. Just lightly rub it 'til it lightens up.
Also, if you want to really put the finishing touch to this consider using some Johnson's Wax. It's the same stuff you would use on a hardwood floor. Just rub it on.... let it dry 10 or so minutes and shine it. You can do this several times if you like. Every time you apply a coat it will look a little better. Also, you can use it to dress up the barrel , breech, etc.
I have surprised myself a few times refurbishing an old rifle or shotgun.
Good luck with your project.
Your son is a very lucky young man to have a Dad that is going to this much trouble.
I commend you.
( Pahoo is likely correct about the birch)