Thanks for all the help -
I've pieced together some parts, and did a bit of trial and error using the original schematic I pictorially showed in the first post.
I have an 240V AC to 18V AC transformer designed for doorbells (10m lead), it actually produces 21.53V AC without load.
I changed the resistor to a 82W 5W resistor, and added a 220uF 25v capacitor in parallel with the coil of the relay and removed the diode at the relay.
The relay is a 24V DC coil, but it triggers from as low as 13V from my tests.
Results:
In normal mode (stable after 10min or so):
-Across the 82 ohm resistor I get 1.15V AC at 13.48mA. If I am not mistaken, that is dissipating 0.0155W (and for a year that would consume 136W (24hrs*365days)
-Across the Google Doorbell (20.68V AC),
When the doorbell is pressed (instantaneous):
-Across the 82 ohm resistor I get OL on my meter (but assume it gets to 21.53V AC and that the auto ranging on my meter is not catching the voltage in time) at 232mA. If I am not mistaken, that is dissipating 4.99W (P=IV)
-Across the Google Doorbell is shorted as expected.
The only addition was a 220uF capacitor in parallel with the coil of the relay, because I think I was getting some odd ripple or something that I noticed when putting a red LED in place with a 1k ohm resistor instead of the relay for testing purposes – it was flickering without the capacitor. I didn’t size up, I just had one lying around so tried it, hopefully I don’t have any inrush issues.
Summary of my tests:
With 100ohm resistor, across the resistor I get 1.60V AC at 15.8mA (222W for a year) and 21.53V AC at 168mA when pressed (3.62W).
With 82ohm resistor, across the resistor I get 1.15V AC at 13.48mA (136W for a year) and 21.53V AC at 232mA when pressed (4.99W) My chosen resistor
With 50ohm resistor, across the resistor I get 0.80V AC at 17.0mA (119W for a year) and 21.53V AC at 376mA when pressed (8.10W). I’d need to buy 10W resistors which were not in stock locally so I didn’t bother
All options sufficiently power the google doorbell battery, and triggers the relay on door press, so I think unless there is anything fundamentally wrong or concerning with my solution (i don't want anything to burn down my house), I can close this out.
On the google doorbell app settings, it has options for ringing an internal chime. With this on, it shorts the contacts momentarily which is the feature I am trying to utilise. It also has an option for ringing an internal electric chime where you can select how long (1-10seconds) it rings for. With this enabled, it just shorts the contacts for the time specified.