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...
You can DIY a debounce circuit or purchase a cheapo debounce circuit board these days for $10. I have done that for my mailbox stuff and it worked well.
Here trigger a door bell ring with the doorbell and front porch PIR sensors plus a CCTV camera such that you do not have to touch the doorbell button or even knock on the door. Ideally a Homeseer TTS/VR AI in a tiny speaker outside would provide an acknowledgement to a person sitting on your porch.
The new electronic doorbells that use the old two wires to the buttons can be easily modded.
The new ones that utilize wireless buttons are not so easily modded.
That is a take apart reverse engineering of the design and a modification to it which can be done if you want.
Only thing is if you spend $200 or more for one of these you are unlikely to modify it for your own use.
There is a discussion here on making or modding one retrofit device to work with your Elk or HAI panel. There is not much to it so it would be suggested to use the one mentioned in the post.
Here the suggestion is to purchase one of el cheapo versions that you see on Ebay, take it apart, learn how it ticks and make it work for you.
Note that many of these are configured to utilize the cloud so it is up to the user how and what he or she wants to do with it.
The question to the OP is:
What specific electronic doorbell was purchased? Model and MFG?