OK, the "Untagged RGB" is now appearing in my status bar. So I made a huge mistake when I created this file 3 years ago! Right now I'm under pressure to complete this piece for an upcoming exhibit, so for now I will have to continue with the "Place Embed" approach, because I'm afraid of what giving this a color tag (which I have no experience/knowledge of) will change some of the hundreds of images I've used. But clearly I need to keep learning for the future so this doesn't happen again. I always just clicked RGB in the Mode menu, and I thought that's all I needed to do.
I don't know what screenshots of the involved images are important to you, because I've worked with literally over 1000 layers in this 98" x 58" file. I've been having this dulling problem only with all digital photos of my painting - except for one, taken by the same photographer. Do you need to see screenshots of all those photos?
OK, the "Untagged RGB" is now appearing in my status bar.
By @Anne5C6B
So then, case closed. Never work with untagged images. If there is no profile, you need to assign one.
What you have now, using an embedded smart object, just sweeps the problem under the carpet for now. You still have an untagged master file, and the problem will return when you rasterize that smart object for final output.
It may look fine, for now, on your system using your monitor, but you have no idea how it will look on other systems using another monitor, let alone when it's sent to print.
The whole point of using icc profiles is that these variables are taken out of the equation. The profile defines the colors. Without a profile, the colors are undefined and random.
So which profile to assign? If you want the file to look as it does now, go into Photoshop's color settings and see what your working RGB is. It will probably be sRGB IEC61966-2.1, or possibly Adobe RGB. Assign that, whichever. That's the profile Photoshop uses internally to display the image when there is no embedded profile.
The problem with the working space is that it does not follow the file, not even in a copy/paste. That's why the profile is embedded. It then travels with the file wherever it goes, defining the color, and ensuring all future conversions are correctly performed.
Once the file has an embedded profile, the working space is no longer important. The embedded profile will always override it. And that's the way it's supposed to work. The working space is just intended as a fallback default.