I think this is a good piece of information for a starting point:
I have follow-up questions:
- What kind of doorbell button were you using before? Was it a regular “dumb” doorbell button?
- Do you have a multimeter available for testing? I’ve used an inexpensive multimeter[1] and set of test leads[2] during my own testing/troubleshooting with my Video Doorbell v2, and having those available has been invaluable.
What you describe seems like the right way to do the standard wiring for the Chime Controller, and you could also try the alternate wiring method (as previously mentioned in this topic) as long as you’re working in the chime box, but at this point I don’t know if your problem is a hardware issue or an app issue.
While it may have added the feature allowing the Duo Cam Doorbell to use the home’s built-in mechanical chime using the Android app, there’s a lot of discussion in the announcement topic about things this app release has broken, so even with doing some testing and trying different wiring combinations, this issue with the built-in chime may require another app update before things work as intended. I’m speculating about this last bit, but if the new app broke some existing things then I don’t think it’s unreasonable to expect that this chime feature might also be broken.
I appreciate your updates here, and I have some ideas of how I’d approach this problem if you have the patience for some long posts and troubleshooting. 
The specific SKU I have no longer seems to be available, but another SKU appears to be nearly identical. ↩︎
These or something similar are handy for clipping to wires, terminals, and multimeter probes while testing and troubleshooting. ↩︎