You state that the wood is birch(hardwood). I didn't think Remington made a Mohawk in hardwood, only walnut, but I certainly could be wrong.
In any event, as long as the scratches are only in the finish, and not in the stain, you could probably just go over the existing finish with Tru-oil and get good results. I've done it many times, as long as there are no deep dents in the wood that can't be steamed out with an iron, and the scratches don't go past the stain, into the lighter hardwood, it should come out ok.
First, I steam out any dents, with a damp rag and an iron overtop of it. Then, lightly sand the finish with 400 grit wet & dry sandpaper, to smooth out any scratches and remove the gloss shine so the Tru-oil has a good surface to adhere to. You don't want to sand down to bare wood, just lightly to give the existing finish a dull look.
Once it is all prepped, take a drop or 2 of Tru-oil, and dab in on the checkering. Work in into the checkeing with an old soft toothbrush. Do this once to twice and let it dry thoroughly till you're satisfied with the checkering. Then you proceed with the rest of the stock. Tru-oil should only be applied with your index finger. You put it on very thin, I call it 'paper thin' and smooth it out with your finger. I usually work it into the area in a circular motion, then smooth it out with my index finger. Work around the checkering, as you've done it already, and don't want the holes filing with tru-oil. You can put as many coats as you want over it till you're satisfied, and at worst case, if you're not happy with it, you can always strip it down to bare wood and start over, with no damage done.
The key to using tru-oil is applying it with your index finger, and 'paper thin' coats, as many times as needed till your satisfied with it.