Your system sounds like it is working correctly. The Soft Proof is correcting the higher contrast of a display image to that of an ink image. But, you prefer the contrast of the relationship between monitor black and monitor white to that of the ink on paper. If you were to print your image to another display, you would be happier than printing the image to paper. If you could get on paper what you see on your display without soft proofing - you would be even happier.
Perhaps, you need to work with an ink and paper combination that has higher dMax and brighter white.
For example, if you are now printing on fine art matte papers you could try printing on Epson Exhibition Fiber. The dMax for example, would shoot to about 2.7+ from 1.64 on JonCone Studio Type 2 (or about 1.48 on Moab, etc.) and the brightness of the paper would be substantially higher than that of a fine art matte paper. Does that make some sense?
What paper are you printing on now? That will give me an idea of where your dMax is.
And also, are you sure you are printing dMax? dMax is only attained at pixel values of 0,0,0 or L0. 1,1,1 or L1 is purposely lighter, etc… might you just need to adjust your files?
What do you think about these options?