[Xringer]

If the capacitive reactance is low, the + pluses will be felt on the - battery terminal.
The + battery terminal will be connected back to the "minus" side of the AC.

It seems like those two voltages would add. Positive peak + battery.
Like to batteries in series.. -+ -+

So maybe, it's not the just a part-time push. If the battery is going to add it's voltage to the whole up-down cycle of each pulse.. No distortion.?.

So, a 12v battery would try to push your 120AC up to 132AC..?.
But, it's limited by the amount of current that can be carried by the cap...