Hm, it even has a "status" output that it sounds like I might be able to plug the telephone mute line into. Here's what the manual says about it:
Status
3.5mm output jack provides 12V DC (100mA Max.) to trigger local devices (i.e. lighting, etc.), where desired,
when line level audio is received at Input B.
This output is provided so that external devices, such as drape pulls, lighting, additional amplifier, etc., can be triggered at the same time as the B Input is triggered. Just be sure that the triggered device will operate with a 12V DC trigger voltage and does not demand more than 100 mA of current draw. Actually, the 12V B Status output voltage will drop to about 9V at the max. current of 100 mA.
So that should work without any need for pull-up/pull-down resistors, right?