The inverter itself should shut down so that it does not backfeed AC to the DB when the mains is off, although as mentioned one should not rely on that. Because its output neutral is not referenced to earth (until it is tied to the DNO supply) the SP MCB does not make it 100% safe if it did start up, as a difference in leakage could make the N more live than the L.
But, the main point of the '2 supplies' stickers around the inverter is to remind that with the AC off, the DC from the panels is still live unless it's dark. So before working on the inverter, both the DC and AC sides need to be isolated.