You can use adhesive cable management tubes to stick to the wall and they'll hold the cables in place and look a little neater. I think Ikea sells them but you can likely find them anywhere that has that kind of stuff

What they can't see won't hurt them. You can patch holes before that move out visit.

Your only option then is surface mounting, really. Drill holes through the walls to make it as short and well/hidden as you can. You may be able to take advantage of the spaces between floor/ceiling joists in one axis only. You can also try to hide it behind cabinets.

Im not saying its a good idea but uuuh