Guillame,
I'm confused by your last post. I don't understand what you are comparing. Are you comparing the original image on screen to the softproofed image on screen before you make any adjustments? And are you doing this with ICC profiles specific to your printer and each of the papers?
I've tested a large number of coated papers using a standard test print,
http://www.jirvana.com/printer_tests...Image_V002.zip, and the manufacturer's ICC profiles for my printer. Apart from the fact that the base stock varies on the cold-warm continuum, most of the papers were quite similar in terms of color. (The Hahnemuehla and Canson papers you mentioned are among those I tested.) I can recall only one that was substantially different, which I suspect was a problem with the ICC profile. Apart from that one paper, the differences were in some cases visible, but they weren't extreme. And consistent with that, the softproof images, without adjustment, were not dramatically different from the original.
So this leads me to guess (only guess) that one of two things is true: the "RC Brilliant for pigment inks" is an odd paper, or the ICC profile isn't correct for the combination of that paper and your particular printer.
However, I may be misunderstading what you did.