Feasibility of Relocating Wired Doorbell Switch on Door Frame to Install Ring Pro

I am considering installing a second Ring Video Doorbell Pro outside my mudroom door. The current switch is mounted on the medial surface of the side jamb as seen in the photos. Instead of using a 30- or 45-degree wedge and creating a bulky affair protruding into the entryway, I wanted to know what I would encounter in attempting to move the wires powering the switch to the anterior surface of the jamb (blue “X”) so that the ring doorbell is forward-facing. This would positively impact function and appearance. Thanks in advance.