I would have to see the internals of the charger, but typical of modern inexpensive electronics
, they probably use an SCR or equivalent to control output voltage. The sharp square wave of a non sine wave inverter is not able to be controlled this way, leading the charger to misread charging
voltage and under or overcharging battery.
There was another thread in which charging options from a 12v battery bank was discussed.
I often charge my battery tools, from a pure sine wave inverter no problems.
If you have 12v tools, or can take a 30% power drop on an 18v tool, just make a cord directly to your house bank.