I am going to sound like a very negative party pooper here but I wonder how much store you can put by the white background fogging explanation. I have a matt white background behind my enlarger( Durst 605) and have never seen the effect you have of getting no change in contrast.
Set up everything in advance then switch all lights off. Put a sheet into an easel or just under the enlarger then expose sections or use two sheets of paper if easier at two extremes of contrast and process( you could even do the processing in darkness until the fix stage) and see if you obtain a difference.
If this cures the problem then all I can say is that you must have an incredible light reflection from the background.
Yes I know a lot of books recommend painting the background behind the enlarger matt black and if the problem is reflection then this should work but I do have doubts if your problem is as bad as you say.
I take it that you have eliminated light leaks from the enlarger actually shining directly onto the paper? When you switch on the enlarger look under the lens. Are there any light leaks from where the lens screws into the plate attached to the bellows?
It is the complete inability to get any contrast differentiation at all that I cannot figure out