GuyNamedErick: The PC Gaming Wiki has a download link to the Joker and Prey in the Darkness DLC files that were from the Mac OS X version of Arkham Asylum, so you'll need to download those files first and then extract them to the BmGame directory.
After that, you can either use the xdelta patch to create a modified ShippingPC-BmGame.exe to let you play as Joker and the other exclusive DLC. Alternatively, you can use a hex editor like HxD and do this:
Search for: 25 00 00 08 00 31 86 80
Replace with: 0D 00 00 08 00 89 86 80
I can verify the hex editing method works for me, not sure about the xdelta patch method though. That said, the Joker and Prey in the Darkness DLCs can be played in the GOG version just fine there.
Article on modding the PS3 DLCs can be found here (Removing : somehow fixes the link on GOG, how broken):
https://www.pcgamingwiki.com/wiki/Batman_Arkham_Asylum#Ex-exclusive_PS3_DLCs How Did you patch teh gog version if the string "25 00 00 08 00 31 86 80" is not there?