Your question is a mixture of three sets of variables: Rendering Intent, Black Point Compensation (BPC) and choice of paper type. It's hard to tell whether your problem is due to your profiles or your management of the soft-proof and print options unless you disaggregate your description of what happens under each set of conditions which you can select. For soft-proofing, I'll throw-in a fourth important one: "Simulate Paper Color" (SPC). Use your soft-proof function to carefully observe what happens with each paper, rendering intent and choice of BPC and SPC, then provide here a careful description of what you see under each set of choices, and then I can tell you whether your observations are consistent with mine under the same conditions. That would tell you whether the issue is settings or profiles.
Thanx for your comments.
This is an example of the most extreme case.
Attached is a comparison of 2 screengrabs (with some jpeg artifacts of course ) in one image showing the softproof settings and a test image containing black and dark grey gradient. Sourcespace was sRGB for this comparison. Lefthand readings in colorsampler window is actual color (sRGB 0,0,0) and righthand reading is proof color. Canon's cheap matte photopaper and their canned profile for this paper, CS2, osx 10.4.11, screen profiled with Optix XRpro.
The problem is not that softproof is inaccurate, on the contrary, even with a cheap iP5300 printer I get quite consistent screen-to-print-match. The problem is (especially with this paper-profile-combination) that peceptual rendering induces some unwanted lightening of the image and some hue shifts. On the other hand relative rendering w bpc does not allow maximum black on this paper (see the difference in the attached image). Now, I can live with this situation because I don't print anything meaningful with this printer-paper-combination anyway but I just noticed similar behavior (although to an almost negligible extent) with a custom profile made for Eps7800.
Question is more about the way black or maximum density is normally achieved in a given profile.