Turn off section fill and you should be able to select what you want and move/edit as much as you like.
Actually I just checked and section fill doesn’t get in the way.
Just make sure you select the parts carefully.
Strange, after I have read the 2nd part of your message, that it even works to select stuff while having the section fill turned on, it started to work suddenly. I swear to god, before it didn’t! I probably clicked too brutish.
Wow, to be able to move and edit geometry while in section plane view is super useful. I have never done this before even though this is so obviously useful.
I find that if the actual Section Plane display is on, it is difficult to select anything behind it, so I only have it on when placing or moving the planes.