Why and how grounding is needed for Inverters

Hi,

I have a very noisy mains line. When my amp(s) are plugged directly into the wall efficient headphones like Focal Utopia sound like a modem from old days. It sounds like data transfer is happening. I tried diagnosing the problem by demoing smaller inverters like ones from Duracell(500Wh) and Ecoflow Delta(1200Wh). They did alleviate the problem, but the issue with them is when in charging mode their fan is constantly on. They are not really loud but distracting.

I am thinking of this particular Renogy Model inverter . My power requirements are really meager - 50W for my headphone amp and 75W for my Holo May Dac. Add maybe another 100-150W for future. So the inverter is kind of overkill but seems cheaper than most audiophile solution. When listening I'll run off the battery and other times it will be charging from the wall.

What I am confused about is grounding in that particular inverter. I am planning on keeping the inverter in my listening room in some corner. There are no children so no one should be going and touching it.  What I am not sure about - is grounding needed in this inverter ? If so how do I do it?  My mental model is - Use the inverter/charger + 12V LiFe battery + DC breaker + DC switch + 3 Pin cable to connect to the charger to wall plug + Modify a power strip and hook that directly to the inverter. Then plug in my Amp and Dac to the power strip. Is there any further explicit grounding needed? If so, where and why?

Thanks.