Just some logical speculation from all I’ve read and seen with my Ring Pro, but I also see a couple of seconds delay between pressing the button on the Ring Pro and hearing the “ding / dong” from the mechanical chime. I don’t believe this can be avoided as it seems common and is probably the nature of the beast. In order to ring the mechanical chime, a connection of sufficiently low resistence must be made between the “TRANS” and the “FRONT” terminals of the chime and that connection is going to short out the power supply to the Ring Pro. But the Ring Pro also needs power to “wake up” its WiFi connection, start uploading video, send a signal over WiFi to any Ring Chimes you may have and maintain the data stream. It seems that this power is supplied by a small battery in the Ring Pro - not large enough to run it for very long - just long enough to ring the mechanical chime. By delaying the mechanical chime ring for a couple of seconds, the Ring Pro uses the power from the transformer long enough to “get ready” to handle an interuption of its power and run off its battery while it powers the mechanical chime, rings it, and then re-acquires full power.
The Pro Power Kit is an integral and important piece of this process. One thing it appears to do is pass power from the transformer to the Ring Pro without routing it through the mechanical chime solenoids - important because without the Power Kit the solenoids would always be passing power (and might buzz or get warm) AND any power lost by partially energizing the solenoids would reduce the power the Ring Pro needs to run, causing it to drain the small battery all the time, eventually wearing it out and leading to a possible early demize of the Ring Pro itself. I believe this is one reason why Ring says to install the power kit before installing the dorbell itself.
Your installation of a second mechanical chime complicates things a little as I have read where Ring says to only install one Power Kit per transformer. If you removed that second mechanical chime from your installation and added a Ring Chime (probably not needing the Ring Chime Pro) in its place you would basically duplicate what I have done - I use an existing mechnical chime (with the Power Kit Pro) in the front of my house and a Ring Chime in the back of the house (far enough away that I can’t hear the front chime).
This doesn’t solve your issue with the delayed chime I know, and I have a delayed chime from my mechanical doorbell but no delay from the Ring Chime. I think if I wanted to remove that I’d add a second Ring Chime in front and let the Ring Pro activate both Ring Chimes and just ignore the mechanical chime altogether.