EDIT: YOU MAY NEED TO ALSO WORRY ABOUT THE UPDATE REQUIRED TO PLAY THE MOD. ASSUMING EVERYONE HAS THE 131111 PATCH MAY NOT BE THE WAY TO GO.
You need to do a Full Conversion mod. If you don't want to do that, you can do this as a workaround:
Have the AAMFP lantern (and sounds if necessary) in your mod, going like this form of directory, where the AAMFP lantern files are the same (correct) filenames as Amnesia's in the hand_objects/lantern/ folder. Seeing as your mod has it, your directory MUST be like this in your zip, and you would require players to overwrite their ATDD lantern.
/redist
- /custom_stories
- /your_mod
- /models
- /hand_objects
- /lantern
What this does however is remove the ATDD lantern. I would include the original ATDD lantern as well, probably in a backup folder.
If your Custom Story however is a full conversion mod, simply overwrite the lantern like this:
/redist
- /<your_full_conversion_folder_name>
- /models
- /hand_objects
- /lantern