Your Samlex is very similar to the Magnum 3012 hybrid I'm using. I can say that if I turn the batteries switch off it does shut the Magnum down along with everything else in the rig.
Granted I'm fulltime never shutting down and it has been near 2.5 years since actually plugging in to a power pole, just using solar as the charging source living off of lfp power.
When I did plug in I still never turned the inverter off, I just set the Float setting to 13.6v (you could lower if it made you feel more comfortable). I never was plugged in for more than 3 weeks tops. If power was lost in the rv park we were good to go.
I'm guessing that you're think of leaving it plugged in during the summer months when not in use and disconnecting it during winter months (Canada weather).
You'll be able to really tweak the settings to your liking/ requirements, so just lower the settings during your plugged in time you should safe and not overcharge the batteries. They'll just sit at a lower voltage in the back ground while your 120 power passes through the Samlex powering the fridge or whatever else running.
Give an idea of my system. Not drop in, my bms/ cpu is to the left.
