Variable gain amplifier output appears on circuit positive and negative supplies
It's a basic fact of life that the output current produced by an amplifier must come from its power supplies. Since the output current necessarily varies with the signal voltage, so does the power supply current. If this variable current produces too much of a voltage drop for your application, then you need to reduce the impedance of your power supply.

Bypass capacitors are one way to reduce the impedance, especially at higher frequencies. But if that isn't doing it for you, you need to explore other techniques.

Personally, I don't see how a 40 mVp-p variation in power supply voltage is going to materially affect your amplifier's output signal. Do you have any evidence that this is the root cause of any problems you're having?