milsabords. It is good that you are exploring these ideas. I don't wish to discourage you unduly. Please consider the below as constructive advice.
"Use of a gas behind the membrane would not work because of PV=RT."
Mariott's law is not a problem: V, R are fixed values. T is measured, so P can be computed
Let me challenge some of the assumptions above. Since the membrane is by definition flexible, is V really fixed? Is T realy uniform?
My personal philosophy in designing mechanisms and devices is KISS.
The fewer moving parts, (or variables) the better.
The law is the ideal gas law (of which the Boyle-Mariotte law is a special case.)
I do not think you would be building on a stable foundation if one used a trapped gas behind the membrane.