^^This might work.
I could be well off base here as I've never done this, but seem to recall can't use a newer version of MacOS to install an older version, even if just to an external drive. Seem to recall that would need to wipe the Mac, install the OS you want, install to external, then re-upgrade back to what the Mac was at. So wipe, Catalina on Mac, Catalina to external, Ventura back onto Mac.
I could be well off base here as I've never done this, but seem to recall can't use a newer version of MacOS to install an older version, even if just to an external drive. Seem to recall that would need to wipe the Mac, install the OS you want, install to external, then re-upgrade back to what the Mac was at. So wipe, Catalina on Mac, Catalina to external, Ventura back onto Mac.