Connecting Ring Doorbell 3 to my in-home electronic Chime Box

1) When my Ring 3 is connected (hardwired but no batteries), my in-home Chime Box (which is hardwired to Ring3) works perfectly but I have no video cam because I lose internet connection when the batteries are out of the doorbell. 2) When I return the batteries to Ring 3, the video cam works fine and is connected to the internet, but I then have no connection to the in-house Chime Box. 3) When I set up Chime Pro and connect it to Ring 3 and the internet, the doorbell rings at the door, at Chime Pro, and on my phone, but not at the in-home Chime Box. 4) Occasionally, there was enough 'trickle charge' to ring the in-home Chime Box. So, I replaced the old transformer with a new AC Power Transformer with Input 120v 60Hz, Output 16VAC 30VA. When tested, the new transformer is putting out 18 volts. Still, the Chime Box works only when batteries are removed from Ring 3 but again, no video cam, even with the new transformer. I have read that there is a way to connect the three: Ring 3 doorbell, Chime Pro and in-home Chime Box. But no luck. So ... how would that work? Chime Pro talks to Ring 3 and Ring 3 talks to the in-home Chime Box? With enough power, why do I even need Chime Pro? It's the doorbell that talks to the in-home Chime Box. The only unknown here is the make of my in-home Chime Box. There's no name on it but I'd be surprised if it isn't a major brand. We've been working on this for days. Can someone please help? Thanks.