Good observation, but maybe the difference is the presence of the dialtone on the good line.
Being a digital electrical engineer who likes to design complicated solutions to simple problems, I'm going to experiment with a circuit that does the following: when the KX-T takes the line off-hook, quickly (in 50 msec or less) break the connection to the CO line and simultaneously put a 500 ohm shunt across the CO line to maintain the off-hook. Wait 600 msec or so for the KX-T to do whatever it does that the CO is interpreting as a dialed digit '1', then reconnect to the CO line and remove the shunt.
Any chance that will work? It will take me a week or so to cobble it together. I'll report back on my success or failure.
