[Floordford]

My first thought when I read the original post was that it might be beneficial to run two pairs of pistons. The two for power and two for pumping air. Also bore the two air pump cylinders out, port and polish the air pump intake and exhaust parts of the head, use bigger valves on the pump parts, and go with a bigger cam profile for the pump. This way air wont move neutral from the air pump to the power making cylinders.

But im guessing there might be extra load on those power cylinders because this is happening, assuming they are in opposing cycles and
Cylinder 1 is an air pump and Cylinder 2 is power making.

C1-pushes air/C2-draws pump air
C1-pulls in fresh air/C2-compresses fuel air mix
C1-compresses air into the intake/ C2-power stroke
C1-pulls in fresh air/ C2-exhausts burnt mix
C1-pushes air/C2-draws pump air

Cylinder 2 is having to now compress 2 cylinders of air instead of one.

And a problem may occur when cylinder 2 opens to get pressurized air. That pressure is going to go back towards cylinder 1 as the exhaust opens again. And because cylinder 1 will be at top dead bottom that air will go there first. Granted im talking hundredths of a second here but that might mess with the way fuel is delivered to cylinder 2. A large surge tank style intake might help remedy that.