You ask about the inverter but never answer your question. I think that it will be very hard to find a commercial inverter that is not noisy simply because all loads other than high end audio don’t require a high quality inverter. And commercail audio is fine with bad power. So why would a manuafacturers waste money on it? The original inverters sold to drive skill saws from a truck battery were actually a square wave.
So I would confirm that I had an audiophile quality inverter before I spent a dime on anything else. The battery doesn’t matter at all for sound, it is all the practical concerns you mentioned.
And I agree you should ovesize the inverter, partly for quiet as you mention but also to esure you have the high power that quality audio needs.
I’ve offered for anyone with the popular battery backups being bought to bring one over and let my PS Audio Power Plant analyse it as an input. I get about 4% thd from the grid. I haven’t been able to test any inverters.
Frankly, I thing the major cost of the PSAudio power plants is the high quality inverter.
Jerry