I've seen your work (read that whole thread this afternoon)
WiFi doesn't cut it at all, but these seemed to have a 433MHz transmitter in them, and you could just get one of the replacement wireless battery chimes for it.
I have half a mind to take apart one of the doorbell buttons and see if...
In my searching around today, I came across a listing that had a PoE powered doorbell and it listed a set of "Digital wireless doorbells" as compatible. Including some from Home Depot.
Stupid me closed the tab without thinking.
Does anyone know what I'm talking about?
Alternatively...