If the switch is in either the #1 or #2 position, the parallel tie between the two 12v batteries is severed. If the switch is in the #1&2 position, the parallel link is maintained. It's no different than switching to "both" in order to supplement a low soc boat engine starter battery with the on-board house battery.
In all three cases, whether switched to #1, #2, or #1&2, power is available to the meter, just as an inverter would be if connected to the common stud on the switch.
I'm not sure that if one battery was "carrying" the other a little, it would take time to see that. I have seen my batteries hold different voltages, even when wired together. And if there was a rather large discrepancy in voltages between the two, the stronger battery would likely draw the weaker battery right down.