Well, not to get technical, but gain and saturation are not the same as they’re classically defined, particularly when you make the distinction between linear and nonlinear behavior 
Gain is generally thought of as a linear commodity, meaning you put a signal in at one frequency and you get the same frequency content out. The ratio of signal levels is of course gain.
When saturation occurs, the linear gain actually drops, but some power shows up at harmonic frequencies.
If we were to make the electrical model closer to the gas/carbonation scenario, the ability of the liquid to hold carbonation would be represented as an RC element with a very long time constant (very high value R and C). Turn up the pressure really high for a short period of time and the amount of gas diffused in the liquid over a given time interval is limited by the resistor. Turn it up long-term, and the gas will become fully diffused to the same pressure (voltage 
) as the gauge setting.
IF, however, you shake the keg to diffuse the gas quickly, I guess you’re kinda shorting or reducing the resistance to diffusion in the liquid, and your cap can be more quickly charged to the input supply voltage
I guess it looks like a capacitive divider, though, where the two caps are the liquid and headspace volume. That ratio and the pressure in the headspace determines what the pressure will be if you turn off the gas and it equalizes over time. I think
