OK, I solved it. (RTFM)
The RC_Data_FMS folder is what threw me. you can't just copy the "Files" folder into that folder and go to town. There are additional folders that need to be added. Once I closed the database, moved the folders within the correct structure in RC_Data_FMS, and reopened, all was right with the world.
Thanks to Wim Decorte and Steven Blackwell for their "Remote Containers - the Filemaker Perspective" whitepaper!