Yes, you are missing something here. You are not meant to assign or convert your image to the print profile. You should be working in a wide gamut profile like ProPhotoRGB or, at least, AdobeRGB. Use the Proofing Setup to allow you to see how the final print profile will affect the image.
Your workflow for a camera should be to convert from the camera profile to something like ProPhotoRGB and work on the image, converting to and from LAB mode as necessary, but using the Proofing Setup to apply the printing profile.
Scanning is similar but, usually, you assign the scanner profile after opening the image in Photoshop but prior to converting to ProPhotoRGB. Then you must not convert to your printer profile, simply set the Proofing View.
You are getting fluctuating results on printing because you need to proof and print using a separate profile for each paper.
Which is why I suggest using PRoPhotoRGB as your default working profile, it is also a much wider gamut.
It is not converting to a profile that causes the contrast change, it is
assigning a profile that causes the visual change but does not change the file.
Then you need to listen to what is being said
Definitive workflow; try this at least once before you do anything else and let us know what happens:
1. Set your default working colour space to ProPhotoRGB
2. Use Image RAW to open an image from your camera.
3. If the import didn't convert to ProPhotoRGB, Convert to ProPhotoRGB.
4. On the View menu, choose Proof Setup|Custom… and choose your paper profile.
5. On the View menu select the Proof Colors menu item ticked
6. Make any edits that you don't want to do in LAB mode.
7. Convert to LAB mode.
8. Make any edits in LAB mode.
9. Convert back to RGB mode (the profile should, once more, be ProPhotoRGB.
10. When you are satisfied with your image, Print it ensuring that the Print dialog shows the Color Handling as "Photoshop Manages Colors" and select your printer profile.
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