Electronics that shuts down when a high load is placed on the battery (such as operating the starter) is an indication that the battery voltage has dropped below the threshold for the electronic device. The reason it does that is because the battery (#1 in your case) is either getting weak or is has too little capacity. This doesn't happen on battery #2 because its capacity and condition are better than battery #1. Put a voltmeter on battery #1 while you crank the engine and I think you will see it drops lower than it does on battery #2.
