I am unable to view your “mockup circuit diagram” but if your Ring Doorbel Pro came with a Bell Transformer and a Pro Power Kit V2 in the box like mine, here’s how I wired it up into my mechanical doorbell.
Since I could not find a spare DIN slot in the Mains Distribution box to install the Bell transformer, I installed it along with the Pro Power Kit V2 inside the existing mechanical doorbell enclosure. I removed all the hardware inside the enclosure and fixed the Bell transformer and the Pro Power Kit.
This, in my opinion, was the most easiest, neatest and convenient way to do this.
After a few months the Bell transformer from Ring which was humming from day 1 and heating up, stopped working. Although Ring sent out a replacement immediately, I bought a BG CUB1 Bell transformer from Screwfix and installed it in the mechanical doorbell enclosure. The Bell transformer is mounted sideways and upside down in the mechanical doorbell enclosure.
Hope this helps.