Your 500 f4 isn't a DX lens. I Thought we were talking about using an FX sensor with half of its pixels. No need to use the FX camera in DX mode. I fail to see how it is relevant to my question. I don't doubt you had a winner in your test but I would suggest I would probably view the difference as trivial as well.
That's not at all what I said in the OP. The question is whether to use my fx lens on my D610 with a TC or whether I put my fx lens on a 34mp Dx body.
*24mp (I think that was just a typo). I think a glass upgrade is better than a D500 though.
When I purchased my D750, I did so knowing one of my main lenses is a 150-600. The upside to a full frame sensor on something like a 150-600 is that you get significantly better pictures when you don't need to crop; when you do need to crop, it's almost a wash between FX and DX, since the lens just isn't good enough at 600mm to take advantage of a DX sensor's pixel density. A side-by-side comparison will show an extremely slight advantage for a DX sensor on a crop. The bigger advantage of a D500 on a 150-600 is the focusing system.
Where DX and teleconverters shine is the tele primes. I think you'd have to spend quite a bit of money to really truly benefit from a switch to DX. An 'older' 500mm f4 from Nikon comes in at $6400 on
B&H 
(which is 'only' a little over twice the price of a D500).