log command (!)
Since the process is running in a screen session already it's just a matter of telling screen to log the output of that window:
Switch to the script's window, C-a H to log.
Now you can :
$ tail -f screenlog.2 | grep whatever
From screen's man page:
Start/stop writing output of the current window to a file "screenlog.n" in the window's default directory, where n is the number of the current window. This filename can be changed with the 'logfile' command. If no parameter is given, the state of logging is toggled. The session log is appended to the previous contents of the file if it already exists. The current contents and the contents of the scrollback history are not included in the session log. Default is 'off'.