It turns out that I mis-remembered how the circuit was arranged. That's what I get when I go months between having an idea for a project and actually following through on it
The button is indeed supplied by the door chime box, and it has 16V (20V unloaded) being fed directly to it. This simplifies my project as now I won't have to worry about switching on 16V current or
maintaining a low-power project running on batteries!
So now my plan is to just run the project by that 16VAC and switch on the doorbell button.
On running from 16VAC - I should be able to use my 50V bridge rectifier diode to an electrolytic cap, then a small ceramic cap to smooth output, then into the arduino's 5V reg should work to power the device (provided the transformer is high enough current), right? Maybe 2 diodes in series to drop the voltage closer to the linear reg's safe area?