well this might work...
unfortunatelly the problem is that in your design the check valve would trap air and the piston/aircylinder would move but you would need a seperate valve to release the air...
note that you don't know if it would work in the first place yet you have to add an additional part just to release air
so to sum it up, you would probably need a well made piston/air cylinder to make this work as well as a check valve and a valve for releasing air to let the piston/aircylinder retract and
you still need a firing valve (+optionally a pilot valve if you want to use a QEV or piston valve)
so 4 to 5 parts
in my design
(note - I don't claim it's mine.. let say it's a collective work
) you need less parts... you only need a 3 way valve, an air cylinder and a QEV/piston valve
so only 3 parts
what's the point in taking air from the barrel when you can use air from the chamber without lowering performance ??
no offence I am not criticizing you... in fact we've all been where you are now... quite a lot of people wanted to build a design similar to yours but AFAIK no one has ever succeded but there were several guns based on 'my' design