OK, I think I'm sorted now. If it helps anyone in the same predicament, the BCM is that thing like a fuse box behind the glovebox on the left. It has about a dozen fuses in it and a variety of connectors plugged into it.
Fiat seem to have protected it from marauding DIY mechanics by putting it behind a metal shield, which is held in with two shear bolts at the bottom and a large pop rivet at the top. I had to drill the rivet out, but fortunately, the shear bolts came out easily enough with Mole grips. After that, there are three set screws (with male Torx heads, so you'll need Torx sockets), holding the box into the car. It comes out easily enough and all the connectors are unique, so you can't get them mixed up. There are also a couple on the back of the unit.
When replacing the shield, I just fitted ordinary M6 screws where the shear bolts were, and an M6 nut and bolt at the top where the rivet was.
