I may be wrong and I apologize in advance if I’m confusing anyone, but the inverter does not have to be on in order to charge the battery. While the ProMariner charger/inverter (on my 2016 boat) is a single physical unit, the charger and inverter functions are separate. The small rocker-type control switch for the unit is on the left as you enter the cave and has three positions. Up is for charging while on shore power or generator, middle is to turn the inverter on and down turns it all off. I leave it in the up (charge) position most of the time - indicated by a green led next to it - and select the middle position (inverter on - orange led) when needed. You cannot be charging and inverting at the same time.
This is all separate from the large rotary AC power selector switch on the AC breaker panel, that, as has been pointed out, simply controls the source of the AC power (shore or generator). The AC1 breaker on that AC panel needs to be on with either choice for battery charging to occur, along with the up position selected on the charger/inverter control switch.
mjq1987 is also correct that there is no DC breaker panel (on newer boats - there used to be one on the older ones). Big red House battery switch in the cockpit controls (almost) all DC power to the various DC circuits, each of which is protected by an inline fuse.
I hope this helps and, please, correct any mistakes. Thanks.