Whats the chances of having a very small diameter piston with long stroke up front which is propelled by the contemporary scissors linkage.
Hidden in the buttstock is another piston of greater swept volume which will feed the small pistoned pump. This is a two stage pump once removed.
Now the cylinder in the buttstock will most likely not end up inline with the forward piston.
So the problem is linking the buttstock piston to the forward piston with some sort of linkage that is flexible.
For this I might suggest a push/pull throttle cable assembly. Which would easily pull the large piston yet have enough rigidity to push the large piston for air replenishing.
Now the throttle cable could be so designed so that the air from the large pistoned cylinder is transported to the forward pump thru the sheathing of the throttle cable which, at the small cylinder, is fastened to a fitting which was o-ringed so as to direct the air into the small cylinder yet allow the throttle wire to be fastened(in some way) to the forward pistons linkage.