Thanks for the additional information! 
I imagine that was probably rated for 16 V AC, 10 VA output, as I also measure in excess of 18 V from my transformer. The new transformer may have been unnecessary, but I don’t think it’ll hurt anything.
Does this mean that for Video Doorbell Pro previously (and now also with Duo Cam Doorbell) you used the Jump Wire installed in your chime box (like if your home has a built-in mechanical chime) to provide constant power to the doorbell camera’s wiring location? I’m trying to envision your wiring paths and what might be happening here.
What has Support said when you contacted them? The experience you’ve described so far is making me think of topics like Hardwire issues with duo cam doorbell, where a user who directly wired Duo Cam Doorbell to a transformer (no chime box in between) was still seeing a battery drain and not charge as expected.