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...