Really appreciate all the advice given above, but none of the solutions worked for me with a 2nd Gen Ring doorbell and a Honeywell DW915S.
I ended up getting a simple 12V AC electromechanical relay, and then adjusted the Ring using the app to ring a mechanical chime. Works extremely reliably, and I can power both the Ring and the Honeywell off the same 16v AC transformer. Hope this helps someone, as it’s much better than relying on the Chime that Ring gave me (which needs an internet connection to function, not just WiFi) and costs £10.