If I understand your question correctly, I think that we can simplify what you are asking to the following:
"I would like to produce a JPG image that looks like what I see when I soft proof a "starting" image."
In other words, we can ignore all the details of exactly how you obtained the "starting" image (ie, ignore the contrast enhancement and any other changes you might have made to an even earlier version of the "starting" image).
If that is correct, I can think of two ways to do this.
1) The first method is to simply take a screen grab with proof-colors turned ON. Some screen-grab software tags the resulting image with your particular monitor profile instead of converting the image to a standard color space like sRGB. If this is the case, you will have to do a convert-to-sRGB_profile step on all of your screen grabs.
Obviously, this method has some big limitations, the most obvious being the maximum resolution of the resulting image can only be as large as your screen dimensions (in pixels).
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2) The second method involves a couple of color space conversions. It removes the resolution limitation of the previous method, but the colors will not be exact because of loss of gamut in the 1st conversion. Specifically, you should be able to come close by using: Edit / Convert-to-profile and selecting the paper profile, and then immediately following that step by a second, almost identical convert-to-profile step, except the 2nd time you will be converting back to sRGB. This process won't be perfect because the convert-to-the_paper_profile is not reversible (ie, you lose info in that step), but it should be close.
Here's an example.
I started out with this snapshot:
To make any changes obvious, I selected a profile, SFprofR(Perfection V700), that I happen to have on my system that I know will impart a strong and weird color cast, as well as make noticeable changes in the tonality of the image. (BTW, yes, I know this is a scanner profile, not a printer/paper profile, but that doesen't matter. All I really wanted was a big, obvious effect.)
This is what I see on my screen with proof colors turned on:
Next, I turned proof-colors off, used the convert-to-profile command and selected the same profile as used for proofing. As the final step, I then immediately used the convert-to-profile command a second time, but this time I selected sRGB. Since the image is now in sRGB, it looks exactly the same when I output it in the normal way using a "Save as" or "Save for web" command. However, for consistency of presentation (all of the previous shots were screen grabs), here is what it looks like on my screen. As you can see, it looks reasonably close to the previous image (ie, screen grab with proof colors turned on). As expected, because of the irreversible loss of gamut in the 1st step, it isn't exact, but maybe it's close enough for your uses.
HTH,
Tom M