This is what I did back when I was first learning about printer profiles, monitor profiles and such.
Since ICC profiles are made to a D50 Illuminant, I profiled my monitor to D50. And for convenience in comparing colors, I set the monitor white point to 100 Nits (cd/m^2). Then, when reading the emissive color with an I1 Pro spectro, one should get a good match between the LAB color in Photoshop with the picker info tool and the spectro reading. This setup makes it easy to measure, not just "see," what happens when you soft proof.
When I put a white (255,255,255) patch on the monitor and measure it with a spectro I get pretty close to LAB 100,0,0. Then, if I select show paper white the luminance drops and the LAB reading drops as well to the paper white described in the paper profile.
If one has a "perfect" printer paper that reflects 100% of the light, and is therefore neutral by definition, then the LAB reflectance reading would be 100,0,0. Show paper white would result in no change at all! No paper does this though some PTFE based proofing papers come close. Most papers like Epson Ultra Premium Glossy (similar to Costco's) are about 85% reflective with a lot of OBA and is LAB 95,1,-9 (M0) The closest I have to a "perfect" paper is a Baryta paper that is LAB 97,0,0 (M0) which is about 91% reflective. The Baryta paper shows less brightness reduction with "show paper white" and, of course, physically looks brighter next to the Costco.
OBAs are another factor. In the case of the OBA laden Costco glossy, using a profile made with M0 (uV but about 30% less than what true D50 contains) the LAB value measured on the screen drops to 95,1,-9. If I just read the reflectance LAB value directly off the paper then I also get LAB 95,1,-9. That's a pretty significant blue shift.
But, I don't actually see that kind of blue shift when comparing Baryta with Costco Glossy. Unless I take the papers outside and view them in daylight. Then I get the blue shift seen when selecting "show paper white" with the Costco but no blue shift with the Baryta. They look vastly different. The reason is simple. My proofing setup currently uses a good quality LED illuminant and has no significant uV. For that reason I make profiles using M2 (uV cut) unless making prints for displaying outside.