Soft proofing seems like a kludge. It seems like it would be better if the ICC profile were applied o the edit screen. What good does it to to edit and then when you soft proof, the colors are different.
IF you only wanted to print that file on one printer with one paper, you could edit in proof preview mode. But that's kinda, like LIMITING. The whole idea is that you process an image to your liking. Then you tell Photoshop to "emulate" how it would look going to a specific printer, using a specific paper. The paper's gamut, and "paper white" impact how your print will look if you "do nothing" to your nice screen image. Soft proofing is a way to tweak the image so that is looks "proper" on "that printer with that paper" while still preserving your original file - just in case you want to send it to another output device like a phone or tablet (sRGB) or perhaps even make a print on a "different printer and/or paper."
Here's a personal perspective. Why in the heck would you soft proof and print out of Photoshop when you can do so from Lightroom, which to my way of thinking is much easier, takes up much less disk space for the virtual copy proof, and where Lightroom does a dandy job of upsampling on the fly for whatever image size you're laying out on whatever size paper you're using - in addition to allowing you to create print "Templates" that store printer driver settings (e.g. having color management turned off) paper size, type, layout, printer / ICC profile, media type, etc., all with the press of a single "click?"
All this "especially" since your image started in Lightroom in the first place. I "tune" most global adjustments (and some local adjustments) in LR, then send off to Photoshop as a smart object (so that I can launch ACR if I want to tweak any of my LR adjustments if I want to) for whatever fine tuning and precise local adjustments I may want to do via e.g. curves layers w/ masks, etc. and other tools. I have Photoshop return my file to LR w/ "_master.suffix" rather than "edit" in the file name. From there I go back to LR and soft proof and print.
Give it a shot.
Oh, and also... (and Digital Dog hinted at this), why would you go from a large working color space in LR and potentially dump a bunch of color data by sending to Photoshop as AdobeRGB? Makes no sense. Set your working color space in Photoshop (and ACR) to ProPhoto, then just do an "Open in Photoshop as a Smart Object" (or just "Edit in" Photoshop if you don't want to use a smart object work flow) from within Lightroom.... Keep all that color data.
Rand