Author here. You can move the sqlite database to a separate location by setting a variable, and that should resolve the syncing issues. I do this and have the files synced via Dropbox across my different machines.
IIUC, the SQLite database is just an index, and can be rebuilt from the underlying data which makes this approach possible. Is that right?
Yep, that's right!
I tried storing the db in a shared Syncthing directory, but I had to make sure I only had one copy of Emacs running on any given machine. I of course couldn't do that so I would get conflicts with the db file and have to manually clean it up. Did I miss something?