I'll let the more experienced gun cranks comment on this proposed approach, but I think it would work.
Assuming that your finish is the original French polish and that you have oil in the head of the stock, you might try a combination of warm temps and whiting, after standing the head of the stock in mineral spirits to dissolve some of the embedded oil. The mineral spirits will wick into the fibers by capillary action, and will not swell the grain. Might take several cycles, but mineral spirits should not damage the existing finish. I would not apply any new finish until I had gotten out as much of the oil as possible. From what I can see in the pics, it almost looks like someone tried to clean off the action with alcohol, and got some on the stock.
