When you don't have hollow walls, the easiest thing to do is make a hollow wall. Or as you've pointed out, a false wall or panel.

Depending on how you want it to look, you might see if you can get gypsum board (drywall) as it will probably take paint easier and you'll be able to match your existing walls. Or you might want to use wood if you want it to look like part of a cabinet or be more decorative. Or a compromise by using thinner hard-board (MDF/HDF) which is very paintable and perfectly flat (no grain).

In you wanted it to be permanently affixed, you would pre-attach some conduit to the panel to make it easy to run the cables.

What I've done for my cabinet and AV install is to both run cables through the cabinet and into the wall where they then drop to the basement and other cables drop down under/behind the cabinet into a space created specifically to channel cables.

If you can post images it would be easier to offer more specific suggestions in terms of how to construct or frame the false panel and how it might open/close if needed.