I ended up with an ESP-32 module connected to the house wifi, and an opto-isolated circuit to trigger it from the doorbell wiring. It sits in the basement where it's easy to give it power from a USB power adaptor.
Of course, the irony was that it was only after I'd lost a few days going down this particular rabbit hole that I discovered that almost every visitor (and especially delivery drivers) uses our door _knocker_, rather than the doorbell since it's much more obvious! The next project may be to electrify that, or mount a vibration sensor on the door!