Bevel an inset face with resulting clean edges When I bevel this Edge (which is parallel to X-Axis)

I always end up like this:

But actually I want this (without the faces that I colored blue)

I extruded the inset face without further scaling and moved it inwards the x-axis After extruding it a second time.
How can I achieve the beveling results in proper edges/faces?
Link to my .blend-file:
https://www.file-upload.net/download-14584958/bevel_problem.blend.html
Here's a somewhat shorter solution, but it still requires a bunch of work.
Create the bevel:

Use the knife tool to replace the missing edges:

Select and delete the edges you don't want, using dissolve edges:

and there you go:

This leaves you with all quads in the replaced area, but a concave quad can be just as bad as an n-gon.