[Setup With Nginx Proxy]

My workaround for proxying a subfolder to BitTorrent Sync:

        location /MY_SYNC_PATH/ {
                proxy_pass http://127.0.0.1:8004/;
                proxy_redirect / /MY_SYNC_PATH/;
                subs_filter_types text/javascript;
                subs_filter 'urlBase=""' 'urlBase="/MY_SYNC_PATH"' o;
        }

i.e. have nginx rewrite the offending line of JavaScript on the fly.  This might be slightly brittle, if something changes in a future version...