Same way existing systems work. The charger just moves the voltage somewhat up above 12V. Under normal use, the battery drops slightly below 12V. You have the things that draw on the battery wired in parallel with the charger. That’s why I’m saying that the power supply on the device would need to be tolerant enough to handle a bit of a range.
An inverter’s power supply will be able to handle that range, but you’re saying that you don’t want the overhead from that. A DC power supply that’s tolerant of that range will also work.
EDIT: There are also gonna be 12V regulated DC input power supplies that you could externally insert.
searches
DROK DC-DC converter input voltage range is DC 9~36V (12V); output voltage is DC 12V; output current is 5A; output power is 60W.
Conversion Efficiency: >90%
Would have to compare to the existing efficiency of your inverter and AC-to-DC power supply.