That wood, beech, I believe, would not have been used for that rifle design, so it's a bit hard to come up with an authentic finish. However, just staying with the kentucky long rifle motif, aqua fortis would have probably been the stain used. Track of the Wolf sells an aqua fortis reagent, I believe. You may be aware, aqua fortis is nitric acid, which can be a bit tricky to use. Great color, however.
I think cherry is too light for that wood. If you want the dark red I suggest you use Herter's French Red grain filler, whisker the stock and then use a good maple stain, like Laurel Mountain Forge. Finish with tung oil or boiled linseed oil, hand rubbed. If the oil finish is too shiny, cut it with Birchwood Casey Stock Sheen and 0000 steel or brass wool.
Here's my attempt at cherry on that beech wood:
It's just not right for a kentucky long rifle, in my opinion.
By the way, refinishing this gun to get a darker red has been on my list for many years....I'll be encouraged if you finish yours.