I did this some years ag on my 156 after a similar incident.
Swapping the doors over is easy peasy (assuming the 147 is the same). Undo the multipin connector, do something to allow the pins of the hinges to come out (can't remember what but very obvious) and the door lifts off. Reverse process and IF IT FITS the job is done.
My problem was that it wasn't a perfect fit. They are adjusted with a combination of shims behind the hinges plus some movement in the hinges thenselves and I couldn't shift the bolts that hold the hinges to the body even with an impact driver (the sort you hit with a hammer). Eventually I drilled them out (easier than I thought) and used new ones. I'm still not sure if the factory uses an enormous spanner or glues them. Maybe an electric impact driver would work?
Alfa GT JTS Selespeed. Previously 2.0TS Sele Sportwagon, 1.8TS 156