I don't use btsync so some shots in the dark here.
Can you not just edit the /etc/btsync/config.json with the proper IP that should've been added during the installation.
Secondly if /var/lib/btsync doesn't exist (which it sounds like it should) it's possible the installation didn't complete successfully and may have thrown an error (but not a fatal one) when the IP it grabbed was wrong.
If I were you, I'd try removing and reinstalling the module again but ensure that no tunnelling or other is taking place. Barring that, manually edit the IP variable in the install script to match your public facing IP and go from there.