Inverters come in two flavors for this discussion and two more of other discussions. (you already decided on TSW v/s MSW so that's why only two)
Inverter/chargers... Work as you describe, have two sets of 120 volt connections, One IN, one OUT, the IN needs to be at least 20 amp and 12 ga wire.
The 12 volt side is bi-direcitonal.
And Stand alone,, Have only 120 volt OUT, 12 volt is input only.
So, your answer is "Perhaps" depending on the type as describe above.
NOTE: I have a 2,000 watt true sine with charger and pass through (Bacically what you described but bigger).. The Factory converter on my RV was a Progressive Dynamics 9180 with wizard... I like the Wizard better than the Prosine for charging so that's what I use.. (I simply disable the charger at the control panel).
HOWEVER... The PDI is a plug in model and where the plug is.. Well sometimes I accidently unplug it.. Now usually when I figure it out the battereis are low and it is like 3am (2 days later) and if you think I'm getting dressed, going out and plugging it back in.. You need to re-think.
So I go over to the Prosine's remote. ... Flip the switch, and watch the batteries charge.