I would say you are beset by a strange set of circumstances that are fooling you into thinking the 110AC power is part of the problem!
The 110AC input is so far removed from the steps, I can't see it being a direct cause. AC powers the converter to make 12VDC to charge the coach side and then the BIM passes part of the power to the chassis side to help it stay charged, so any changes in the AC side is seems pretty far off from getting to the step powering!
But with no proof, one way or the other, I would suggest some testing that might lead to some info gained?
One of the little things that can get flaky enough to make us think we are losing our mind is the little magnetic reed switch at the bottom of the door that senses the door position. Know where that is?
That is a really thin little wire "reed" and it can get defective and stick either open or closed at different times. Maybe make sure one of the little wires is not loose on the switch?
Next time things go goofy on the door, try just gently thumping that switch with something like a screwdriver handle to see if it jars it loose to make the door position show correct and that is what is needed to make the steps pop out!
Those steps are a whole lot more complex in their "thinking" than I first expected and I still don't have many of the details down but that switch is one funky little item!
That would be my first thought and if whacking it a bit makes it work, then try a new switch at the door there!
But as second idea, there may be a loose plug or connection on the steps that is getting jarred just enough to make connection when you go in and out to plug or unplug, so maybe banging around on the steps is a good second plan if it fails again?
Maybe stand over the step and stomp your foot?
Heck of a way to do things but when it's intermittent, kicking it can be as good plan as any!
I would guess it "could" also be in the step electronics that sense current flow but that's one I would want to try to make sure it was not the easier stuff before going there!