You said …“There’s no noise if no source is plugged into the amp. I suspect this is because it essentially has a ‘sleep’ mode if no input signal is detected, though.”…
Doesn’t this imply that the ‘noise signal’ is being picked up in the input lead? And does the device supplying the noise signal do so into another amplifier (if you have one around) in the same location?
It’s usually Ground Loop /or/ Mains Power interference, and perhaps the injected noise is from the other device which is reacting to the inverter.
Just a thought
Did you manage to solve the issue?
I have a very similar issue with 13A-plug or 7kw charger, they both work fine with the battery and inverter off and isolated, or with the battery in pause charge/discharge mode, but never with the battery in idle, charge or discharge modes. Its clearly an issue with low frequency noise put into the power line by the inverter, since multiple ferrites on the supply from the inverter with frequency suppression at ~ 1MHz have no effect.
The only option seems to be to add a power line filter to get rid of the added noise in the kHz range.