How do you install a ICC profile in your camera to get a correct exposure?
Sadly, of course, you don't....
ICC profiles are more about getting colors correct.
But exposure is the most fundamental part of getting colors correct! And, in a fully-ICC-managed workflow, the profile is going to correct any exposure errors at the same time it gets the colors in line. A hybrid workflow will at least correct exposure and white balance, which is all you can realistically do outside the studio, leaving it up to your supply of carefully-constructed generic profiles (one in daylight, one in shade, one in tungsten, etc.) to get you "close enough."
Obviously, you need to shoot a target in the same conditions at the same settings as the "real" shot. If you're shooting tethered in the studio, you can quickly build a profile on the fly and analyze it to tell you the exact exposure adjustments and white balance settings you need, and you can then decide if you want to make your exposure adjustments with the camera / lights or (if there're reflective or illuminated highlights that'll get clipped) in post-production. When I'm doing art reproduction, I do so and adjust the lighting to within 1/100 of a stop of perfect exposure in camera.
The same process also (of course) tells you how to adjust exposure and white balance in post-processing, at which point all you're left with is applying the ICC profile.
I use a similar workflow in the field when I'm doing landscape work, except that I generally don't shoot tethered and thus I'll bracket (usually with the auto-HDR because it's so convenient) and pick the best / least-worst exposure as the starting point, and then use the profile of the image of the chart to determine post-processing adjustment of exposure and white balance. Works awesome. Even if you significantly miss the exposure, you've got so much latitude with modern cameras that, so long as you didn't clip the highlights, you can perfectly salvage all but the worst mistrakes...so long as you got that shot of the target (I use a ColorChecker Passport in the field) in the light with the same settings.