I don’t even see in the photos where you’re running wire from the new transformer. How are you routing the wire? I would expect to see wire ends coming through the door frame and into the rectangular hole in the back plate, just below the “UP ⇧” in your second photo.
For my own doorbell installations, I like to clamp and solder spade terminals (the kind that look like small two-tined forks) onto the wire ends so I have a nice flat surface to slip beneath the screw heads on the terminals on the back of the doorbell camera unit. I think that helps to keep the wires tucked out of the way. Since Video Doorbell Pro is supposed to include two extension wires, I would expect those to have spade terminals already attached to one end of each, and the other end could be nutted or spliced to the exposed wire ends from your transformer. You shouldn’t need a lot of wire length poking through the door frame or wall (if that’s how you’re routing the wire), and once it’s connected to the back of the Video Doorbell Pro, you should be able to push any excess back through the hole so that the doorbell camera unit can slip onto the back plate.
That’s how I’d plan to do it, anyway. Without more information (maybe more pictures), I’m having difficulty seeing your problem.