While I see the specs of the charger I do not see an operational setting to enable just the charging system while the incoming power is ignored.
Best shot seems to be inverter-off mode and still the box is looking for 60 Hz so the power will be stopped including power to the charger.
I would trade that transformer in for a local battery charger rated 10 to 50 amps into a 12v battery. Plug the charger into the power source direct and charge the battery direct. For 120v power simply turn the inverter on.
Don't see this working any different way. You may want to contact help on installation/operation as I believe that is the issue of how this operates over needing service/repair.