Smart alternator, dc to ac inverter, ac auxiliary battery charger, will that work?

New guy here, and I have posted intro in new member section.

I have been reading lots of threads about the smart alternator and different ways to charge an auxiliary battery, including about guys who added an additional dc to ac inverter. But I did not find any threads about combining dc to ac inverter with ac AGM battery charger. I understand the main problem with this is loss of efficiency.

I have cab high camper topper, with 8 gauge wire and fuse block added. I bought a 24F Walmart Everstart AGM battery as back up, and if necessary, replace stock battery. I have a 10 year old 750 watt, modified sine wave dc to ac inverter and new 2 amp ac NOCO (Amazon) battery charger. Can I use these to charge the auxiliary battery in the bed, without problems of undercharging from the smart alternator?

I did hook up inverter and battery charger to fuse block in the topper and auxiliary battery was charged with about 120 watts, so that part works. But I did not start the engine to test how the smart alternator would react because I would like to get some verification if this would work from people who know more than I do.

I have read up on Victron, Redarc, Renogy dc to dc converters to charge auxiliary battery. But then I still have to add a 1000 watt dc to ac inverter to be able to use power tools.

I am thinking about getting a new pure sine wave 1000w inverter so that I can disconnect it from auxiliary battery charge circuit and use it directly off either starter or auxiliary 12v battery to be able to run basic power tools.

Regardless of which way I go with this, quick disconnects will be used for all of the equipment connections so I can remove everything from the bed and store it in my garage. I am to old for any more camping, and I am setting this up as an emergency bug out truck and may have to use it only once or twice, hopefully never.