The inverter/charger will power stuff directly just as the converter does. In both cases, they're connected in parallel with the battery, and on average will power the house loads (and reduce the charge rate to the battery) up to their current limitations. Of course, it's not really possible to tell exactly where any given electron came from, whether the battery or the charger/converter, even if you could identify them individually.
It should be a reasonable upgrade, I think. I would consider installing a 12V outlet for the CPAP machine—it would be more efficient than going via an inverter of any sort, and the installation isn't all that difficult. The hardest part is physically running a wire.
By the way, your battery bank is 300 Ah, not 300 mAh—perhaps that's just a typo. That's about 3.6 kWh total energy storage (at a nominal 12V), if those are easier units to work with.