I'm afraid you can be right. The cars now are wheeled computers in fact.
The FIAT airbag module is very sensitive to voltage fluctuations, drops etc. There are many known cases that it needed a specialized repair after playing with the connectors (however mostly of the ones related with the security systems - airbags and pretensionners) or battery. However this might not be the case.
I'd first try to disconnect the battery safely for some time (ignition key out, all cables and connectors in place, all receivers switched off, all doors closed - of course bonnet up, as I assume the battery is in the engine compartment, let the electronic modules shut down, so leave it for a minute or two, then remove the negative terminal from the battery), unless you tried and described in post #3. Reconnection in the same way.
Then, if the warning light does not goes off, I'd suggest connecting to
MultiEcuScan or AlfaOBD, they are able to connect to the airbag module with the required adapter (list of supported modules and necessary equipment on the multiecuscan.net website). And then check for the errors, clear all of them. And recheck
Attached is the eLearn documentation that contains the instrument panel (page 25) and the airbag modules (page 79).
It could be also useful to discover if the LS/LX versions instrument panel could be installed in place of the LE one.