First, I am still struggling to fully understand color profiles, but I feel like I have most of the basic concepts. I welcome any corrections to any misunderstandings revealed by what I am asking, particularly when I make an assertion about what I understand Photoshop to be doing.
I have a number of TIFF images that were embedded with a custom color profile by the scanner that produced them. Some of the images were subsequently edited & saved in a program without any sort of color management enabled, and the profile from scanning was lost. As a result they are clearly "miscolored" when opening them in Photoshop, which I believe just assumes they are in the working color space sRGB because it doesn't have any better information.
When I open some of the images that were left "untouched" after scanning, Photoshop says the source space is the custom profile. If I paste the pixels of the edited images into one of these untouched images, they look much more correct. Assuming for the moment that the untouched images do in fact have the same profile that the edited images are supposed to have, how can I assign that profile to the edited images? I would like to get the edited images back to their original colors, or as close as I can get to that as is possible. The custom profile from the scanner isn't one of the options in the drop-down, and "Other" is greyed out. I tried to find a way to export the profile from the untouched images so that it can appear as one of the "Other" options, but couldn't figure out a way to do that. If it's relevant, the custom profile is "input-only."
In short: How can I save a custom color profile from an existing image to use in the Assign Profile function? Or alternately, how can I apply the color profile of one image to another image without using Assign Profile? I assume that pasting the pixels into one of the images with the correct profile is an option, but that seems like a hassle and is probably fraught with other issues. Thank you!
Edit: As requested, I have attached images with the sort of custom profiles I'm referring to. For simplicity in the question I just referred to a single profile, but I come across all sorts of different ones all the time. I also am working with TIFFs, but it would not let me upload them so these snippets were converted to JPEGs.