Wired Doorbell Pro won't ring existing chime

I am a new Ring user and I'm still feeling my way along. I could use some help/advice.I installed a Wired Doorbell Pro system a few weeks ago, and it seems to be working properly so far. My issue is that it only rings at the doorbell and on my phone. I have been trying to get it to ring my existing mechanical chime, as well. That chime is a Heath/Zenith "35 Base", which is listed as compatible within the Ring documentation. I acquired and added the Doorbell Pro kit to the chime with no change. I just replaced the existing transformer with a more powerful 16V 30A unit, thinking the original did not provide enough juice to the unit. No change here, either.At this point, I seem to be out of ammo. Any help would be appreciated. Thanks in advance!