The hiding from behind I think you can achieve by enabling Backface Culling in viewport shading settings. As for hiding it in a particular viewport…
If it’s an actual plane with texture, you should be able to use “Local Collections” in the sidebar -> View tab, i.e. put the plane into a collection and disable that collection for one specific viewport.
Reference/Background images (i.e. Empties with images on them, ones you add via Add -> Image) have a set of settings that allow to enable/disable perspective/orthographic viewports, and front/back.