You can not just change a body computer and expect it to work.
The body computer has to be in a "virgin" state, that is, not coded to the vehicles key code security module.
As I recall there are 4 components of the Fiat security system.
1) The engine ECU
2) The Body computer
3) The Key control ECU
4) They Keys
The keys are programmed/stored into the Key control ECU
When the vehicle is first power-up during manufacturing then all units are in an un-programmed (virgin) state. On first power-up they exchange credentials and from that point onwards they are "TIED/LOCKED" to each other.
To replace any of the units (Engine ECU, Body Computer, Key Code Unit) then the units have to be in an un-programmed (virgin) non previously mated state.
This is the basic position. I've never had to replace any of these items in a Fiat so the extract requirements and procedures are not familiar to me. This includes the exact procedure of replacing and introducing a new unit into the system.
Maybe others can expand / offer further information.