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.