The game works in Catalina if you disable some of the System Integrity Protection features of the OS - notably, the Filesystem Protections flag.
Code:
hquest@Alexs-iMac ~ % csrutil status
System Integrity Protection status: unknown (Custom Configuration).
Configuration:
Apple Internal: enabled
Kext Signing: enabled
Filesystem Protections: disabled
Debugging Restrictions: enabled
DTrace Restrictions: enabled
NVRAM Protections: enabled
BaseSystem Verification: enabled
This is an unsupported configuration, likely to break in the future and leave your machine in an unknown state.
hquest@Alexs-iMac ~ % ps -ef | grep -i final
501 692 1 0 12:50PM ?? 0:26.39 /Applications/FINAL FANTASY XIV ONLINE.app/Contents/SharedSupport/finalfantasyxiv/lib64/../bin/wineserver
501 698 1 0 12:50PM ?? 0:00.06 /Applications/FINAL FANTASY XIV ONLINE.app/Contents/SharedSupport/finalfantasyxiv/bin/FINAL FANTASY XIV ONLINE C:\windows\system32\services.exe
501 702 1 0 12:50PM ?? 0:00.04 /Applications/FINAL FANTASY XIV ONLINE.app/Contents/SharedSupport/finalfantasyxiv/bin/FINAL FANTASY XIV ONLINE C:\windows\system32\winedevice.exe
501 706 1 0 12:50PM ?? 0:00.04 /Applications/FINAL FANTASY XIV ONLINE.app/Contents/SharedSupport/finalfantasyxiv/bin/FINAL FANTASY XIV ONLINE C:\windows\system32\plugplay.exe
501 710 1 0 12:50PM ?? 0:09.76 /Applications/FINAL FANTASY XIV ONLINE.app/Contents/SharedSupport/finalfantasyxiv/bin/FINAL FANTASY XIV ONLINE C:\windows\system32\winedevice.exe
501 722 1 0 12:50PM ?? 0:00.04 /Applications/FINAL FANTASY XIV ONLINE.app/Contents/SharedSupport/finalfantasyxiv/bin/FINAL FANTASY XIV ONLINE C:\windows\system32\rpcss.exe
501 683 1 0 12:50PM ttys000 0:00.19 /Applications/FINAL FANTASY XIV ONLINE.app/Contents/SharedSupport/finalfantasyxiv/bin/FINAL FANTASY XIV ONLINE winewrapper.exe --enable-alt-loader macdrv --wait-children --workdir C:/Program Files (x86)/SquareEnix/FINAL FANTASY XIV - A Realm Reborn/boot --run -- ffxivboot64.exe
501 714 1 0 12:50PM ttys000 0:00.67 /Applications/FINAL FANTASY XIV ONLINE.app/Contents/SharedSupport/finalfantasyxiv/bin/FINAL FANTASY XIV ONLINE C:\windows\system32\explorer.exe /desktop
501 728 1 0 12:50PM ttys000 41:55.33 /Applications/FINAL FANTASY XIV ONLINE.app/Contents/SharedSupport/finalfantasyxiv/bin/FINAL FANTASY XIV ONLINE C:/Program Files (x86)/SquareEnix/FINAL FANTASY XIV - A Realm Reborn/game/ffxiv_dx11.exe //**sqex(snip)=p**//
501 852 381 0 1:27PM ttys000 0:00.00 grep -i final
However, this is of course an obscure method and not recommended. Do it at your own risk if you cannot go back to a previous OS X version.