There are times when the inverter on the engine battery, powering a battery charger hooked to the house batteries is acceptable, and ta few circumstances when this is preferable to a solenoid.
One of these is when the house batteries are at a great distance from the alternator. Since there is very little voltage drop on AC lines one can put a higher voltage/ current 3 stage charger on the house batteries powered by an inverter attached to engine battery and actually get increased charging current.
Also, Some vehicle voltage regulators are set too low, and combine this with long thin runs of copper to feed the distant house batteries, this method is a viable option.
However one pretty much needs a dedicated inverter very close to the engine battery, then a quality 12 gauge extension cord to feed the grid powered AC battery charging source. The extra heat under the engine compartment will shorten inverter life, water intrusion might kill it, there are 15% inefficiencies minimum on any inverter, pure sine wave or modified square wave inverter, and another 15% inefficiency in the AC/DC battery charger.
Add up the costs of this vs using 2 awg copper, or thicker the total distance between alternator solenoid and house battery and it is difficult to say which is going to be more effective without knowing all the many variables.
I personally prefer thick copper, but there are times my vehicle's voltage regulator decided 13.7v is plenty and recharging current attenuates to a large degree, compared to if the VR was allowing 14.7+ to continue.
Also if going the inverter route, one must keep in mind hot alternators are limited in their output, especially when asked to exceed their capability at slower, idle speed rpms.
If the inverter is large, and the AC/DC charger is powerful, then it is likely the alternator can not keep up, and the engine starting battery would be depleted to charge the house batteries., and modern starting batteries do not like to live their life less than 95% charged and will degrade quickly if cycled deeper than 20%( 80% charged).
So is it possible, yes, will it be cheaper and more effective? In 99% of rigs not towing a trailer, no. If an equal amount of money was spent running thick copper to distant batteries, the copper would likely be more effective, and possibly much much more effective.
But if one already has an inverter, and a quality AC extension cord, and a charger, then this can certainly work, but I would recommend saving for some good copper with proper terminals professionally attached, a high quality solenoid, as well as taking power directly from alternator to feed solenoid and house battery bank.
Taking power from engine battery leaves the original OEM charging circuit as the weak link, and tricks the voltage regulator into thinking the house batteries are more charged than they are, and cut back voltage, which cuts back charging amps.
There are dozens of ways to charge a battery and isolate house from engine batteries with the engine off. There is no one right way. I like extra copper because less can go wrong, if fused properly. I personally employ manual switches, so I dictate where charging current goes to and from, and where discharging current is taken from. But my system would confuse 99% of people at first glance and explanation of how it functions. Sometimes it is not so convenient to go and turn the switch when I want the alternator to feed the house bank, or turn it off so the house battery does not try and assist the starting battery.