I am in UK.
After much investigation I decided to buy myself a Ring Doorbell 2. My existing doorbell is Honeywell DW915 powered by batteries, and I have two wireless plug-in extenders round the house. Ring advertised that their device could operate existing chimes, and did not need external power, which is why I bought it. I then spent several days going round in circles and could not get it to work, and eventually found a compatibility list hidden away in a corner of the RIng web site, listing my chime as incompatible. I phoned Ring, who were useless. They said take it back and get a refund. I can do so and also send back the spare battery pack I have ordered from Amazon, but it’s going to cost me money, and I would rather get it working. They also said I can get Ring plug-in chimes which will connect, but I have £100 invested in my existing equipment, and would have to spend the same again to get the Ring devices. I thought of battery-powered relay to trigger the existing chime, but I can’t find one anywhere. Should I just give up?
Yes, for the project as you describe it.
Instead, if you want to keep the ring for its video capabilities, I would look for a less expensive chime device that could be triggered via SmartThings or Echo, assuming you already have one or both of those.