When I get my Transit soon with the same setup (monitor in mirror) I am going to try to figure out a way to do this.I have a 2016 T150 cargo with the standard radio. The rearview monitor is in the mirror and it is very good. I wish I could also manually turn it on at times to check the wider view the camera provides.
A wiring diagram for the rearview camera system would be extremely helpful. If anyone knows where to find one for free or cheap let me know. I usually use the Alldata system that is available at my public library but the Alldata info for 2015-2016 Transits does not include schematics (yet).
In my current van, a 2013 Chevy Express with rear view camera and monitor in mirror, I was able to get the system to stay on any time the ignition is on. I followed the schematic for that van and found the +12v signal from the reverse lamps that activated the rear camera & monitor. I cut that wire, added a diode inline to prevent the power from backfeeding to the lights and keeping them on all the time. Then I just ran a +12v ignition switched line from the fusebox to the camera side of that wire.
I hope it's that easy on the Transit but I won't know until I have it or I find a schematic.