I’d be concerned that the inverter produces dirtier electricity that the wall outlet. All the marketing managers say "pure sine wave". i’m sure your wall outlet easily meets their requirements. Inverters are not easy.
Second issue that a lot of people don’t think about, maybe you have, is that when your battery is plugged in for charging you’re running off wall power, not battery power, so you have to physically unplug the charger while you are using the battery.