I've managed to deploy a test app from Monterey which works on Catalina and Monterey and given up on building on Catalina after spending a few days trying to fix the result of macdeployqt using otool and install_name_tool.
It seems that the CMake qt_generate_deploy_app_script method doesn't work when run from Monterey either. Running the macdeployqt utility almost works out of the box, it just leaves the plugins' RPATH pointing to a nonexistent 'lib' folder where it expects Qt frameworks to be. Rather than fix the RPATH's in each of those plugin libs, adding a 'lib' symlink pointing to the Frameworks folder seems to do the trick.
In case it's useful to anyone, I've made a template repository on GitHub with this solution. The template allows you to deploy apps with LGPL Qt on Mac and Windows and use VSCode as an IDE rather than QtCreator:
https://github.com/yergin/qt-cmake-vscode