I like the idea of the inverter.
Like I said it all depends on the amp/hrs of the battery, for the job it’s to do, you can charge it with whatever you want, but never have it charging while your listening, defeats the whole purpose.
I have a very nice sounding MSB discrete R2R Dac that has 5 x half brick size gel batteries in it for the different power supplies in it, it charges the batteries when your not listening and disconnects the charger when listening.
Also My Lightspeed Attenuator runs on 12vdc Li-Ion batteries lasts up to 2 weeks before a recharge is needed , most of my customers buy it also as a "purest" power option, as I only supply a linear wall wart with it.
https://www.ebay.com.au/itm/US-Portable-Super-Rechargeable-Li-ion-Battery-Pack-DC-12V-6800mAh-for-CCTV-Cam/153537757150?hash=item23bf903fde:g:CSwAAOSwq7JT5MSO Cheers George
In my experience batteries do not have good current delivery.
Someone needs to go back to school, better still, put a dead short across the terminals of a small 12v SLA battery, but get "real close" as you may not see the current arc. 🤦♂️
https://ibb.co/4ZwDct1