This is a bit of an unanswered question right now! Lots of people are asking for different reasons (eg to power a 12V fridge when camping). But I haven't seen an answer. Test is easy to do!
1) Move the fuse so the 12V socket in the back is powered all the time.
2) Measure Voltage across the 12V battery (and continue to measure throughout the test). Voltage should read 12.8V (or thereabouts).
3) Attach a load, ensure it is continuous, a few hundred Watts is perfect.
4) Monitor Voltage.
The voltage could slump a little as soon as the load is applied (depending how big the load is) eg to 12.6V. As power is drawn from the 12V battery and it becomes discharged the DCtoDC converter will hopefully kick in and Voltage would suddenly rise to 13.8 or 14.4V (or something similar). If this happens you know you're good to go and can use your inverter as you describe without exhausting the 12V battery.
An alternative is to force the DCtoDC charger to stay on by keeping the seatbelt connected and putting a weight on the seat.