It is most probably a binary, that is why you cannot find this.
I managed to add a timestamp with the following command (read the log file, in this example I filter only the connects, and redirect to a new file). You can also redirect to a remote server, the sink as you call it.
tail -f /tmp/karma-phy0.log | grep 'pass\|AP-STA-DISCONNECTED\|Successful' | awk '{ print strftime("%Y-%m-%d %H:%M:%S"), $0; }' >> /sd/karma.log