I've done a little more poking around, and it looks like disabling the check for permissions in the syncthing GUI makes things work. I've gone back and checked, everything I mapped before has switched to actual file extensions and real files, rather than the .tmp garbage it was copying before.
So, if someone else stumbles here in the future (since this is the very first result on google!): hit your syncthing GUI, open each folder, and check that "Ignore Permissions" box!