Hi Pete;
He could indeed use the Elk-960 and trigger with the AC solenoid's contact closures, but that's another $30 or so, plus another thing to install.
I wish these electronic doorbell manufacturers would provide external contact closures with their products. I think if this were me I would at least take apart the unit and poke around with a meter to see where I could pick off a voltage output when the doorbell 'dinged'. You would have to use caution on what you put in there though (hi resistance solenoid for instance or transistor switch) so you don't risk drawing more current than the device was designed for...