I think what technique will work for you has a lot to do with how patient you are. Before there were power tools magnificent furniture (and gunstocks) were made but it took a lot more time and patience. I do not have that kind of patience but I've made a bunch of furniture and a few gunstocks.
I would not attempt a gunstock without at least a plunge router. A router table is better at some things but, as has already been mentioned, if you have a router and a piece of MDF or smooth plywood, or a sink cutout you can make a router table quite easily. The OP seems to be struggling with the barrel channel (on an airgun it is really typically the shroud that fits into the channel). I get a router bit about the right diameter and use my router table. It's one of the easier aspects if you do it this way. Cutting all the areas out for the action to fit into and doing it accurately is harder in my opinion. A plunge router will work for this as will forstener bits on a drill press. I use my domino but that costs as much as a nice air rifle.
I would not try to use a washer as a scraper but maybe it will work better than I think. You could just buy a card scraper and use a grinder to make a scraper that will work. Or you could spend a few dollars on a router and bit and be done with it. People more patient than I am can get a lot done with slower working tools, however.
