I know about this issue, and can reproduce your steps. Read through this topic:
Short version is that the tolerance of what makes a coplanar face is not the same as the tolerance of edges of a face. You can have non coplanar edges that create a coplanar face.
Look for sw_flatten_to_calculated_plane.rbz from @sWilliams in that topic. With that extension you can select your southeast face, use the extension (Flatten to Calculated Plane in the Extensions menu), and the intersect won’t introduce the invisible new edge on that face. It will still on the other faces, so you have to fix them one at a time.
You could use the extension after the intersect, and then only have to fix the faces that will show a problem.