Final Cut Pro 10.4.6 incorrect color in viewer

If it were a ColorSync issue I'd expect it to affect all clips since it's an OS level setting. Why are you changing ColorSync profiles?

An easy thing to try is resetting FCP Settings. Hold Option and Command while launching FCP and choose Delete Settings. It will open the default Untitled library. Re-open the library you were working on, set the Settings back to the way you like them (disable Background rendering, etc.) and see if the issue has been resolved.

If not, try removing all effects from the clip by selecting it and choosing Edit > Remove Attributes (or press Shift-Command-X). See if the clip looks any different. I would expect a problem here to also show up in exports, but you said the export looks correct.

What are the library settings? Standard Gamut?

What are the project settings?

What are the specs of the problem clip? (codec, file type, frame size, camera used, etc.)

Is there a LUT applied? With the clip selected, go to the Inspector and make sure the bottom popup is showing Extended, then scroll through and see if a LUT is applied.

Yeah, Color Sync would affect the whole monitor, not just one pane in an app.