Or, here's an alternative idea, more aggressive:
What if Eglot just sets project-mode-line=t in eglot-managed buffers, and removes the project-name from the Eglot entry entirely?
Then the language identifier would be the major mode, the project identifier would be project-mode-line, and the eglot status indicator would just be for the status of the server.
Works for me, it's in line with Eglot's policy of setting other modes when managing buffers. Show a patch.
But also, my idea of eglot-mode-line-format should be alsoh considered. AFAICT there no big technical hurdle, it's just relatively boring work and some naming decisions to make. And as I mentioned, it would fix more issues than just the one at hand.
João