is it possible to cut and delete paths that are not visible?

hi guys,

i'm trying to find a way (something automated not manual cleaning) of path lines that are hidden behind other objects with fill color. I've attached a simple example of what i'm referring to.

shape A has a single path (no fill) behind a rectangular box with white fill. Pic B shows the actual paths with no fill color.

Now I wish to remove part of the vertical line that is behind the box. So basically pic A should look the same when viewing the outline/paths (with CTRL+Y)

I have tried all pathfinder options and non of them did the job. The only way I know of is to use the shapebuilder tool and manually drag and merge the internal of the box. It works for this example, but I need it for far more complex artwork that need to be cut on laser/sticker cutters so the shape builder wont help me in this case.

I also tried selecting all filled objects, moving them on top and use minus front but unfortunately open paths will get deleted completely. Minus front between 2 closed objects has the behaviour that i'm looking for, but as stated, not working with open paths.

any help much appreciated. I'm open also to 3rd party plugins/software, though i would prefer doing it natively