Yes, it should. The inverter/"charger" name throws a lot of people off, who are more used to the traditional RV setup where a direct connection between the AC system and shorepower is more common.
The way you have it in your conceptual diagram, is to the best of my knowledge how inverter/charger's are designed to be wired. And is not a limitation in the way that many unfamiliar with inverter/chargers assume it will be.
But, there are specifics that may be unique to your model/brand inverter/charger. And there are surely pros/cons to the different approaches that I am unaware of.
I
think Rucker's assumption is that (A) invert, or (

charge the battery when connected to shorepower. But I
think this is mistaken. Considering the scenario they set out (depleted batteries, connected to shorepower), I believe your inverter/charger would be capable of (1) powering AC loads directly from shorepower (passing it through), (2) powering DC loads and charging the battery at the same time up to the limit of the charger (85A @ 12V in your case I think).
I think the misunderstanding is assuming the charger can
only charge the battery. But to the best of my understanding electricity has no awareness of or bias toward where it flows, its like water or wind, it takes the path of least resistance, and just because its called a charger doesn't mean it can only charge the batteries, if there are hungry batteries and hungry loads in the same circuit, the charger will supply power to both, just as the solar charge controller would, despite the name charger in the titles. Its quite possible I am understanding something (of a technical nature, or the implication of Ruckers comment, I'm a little dense sometimes, so set me straight if I've misunderstood anything).