I installed it all in my basement. But I floated it. I put down clear 3mil plastic sheeting the entire concrete floor. Since concrete is pourous material, I didn't want any moisture coming up, even though the flooring says vapor barrier included. It's like $10 for 200 sq ft of plastic, just do it. Taped it in place and put all the flooring down. Never a moisture issue and my slab is a floating slab that lets water run under it to the sump pit.
Also the moment i saw your pictures, i said to myself he's never going to get that level again, you're going to have to redo the whole room, and those planks really are sensitive to off square/off plumb in my experience. I was glad when you said she didn't care if the flooring was not the same height in that room, saves you a lot of hassle.