I'm pretty sure that the Stilos use the same airbag systems as the Alfa's I've experienced previously - the allied MY99 (maybe MY97 in earlier cars).
You'll find with these ECUs that they don't clear the fault light if the fault disappears like the engine ECU might. The engine ECU if it throws a fault light will put the engine light on, but if after around seven engine starts the fault has not been present it'll remove the warning light but keep the code stored.
The airbag ECU doesn't do this - fault detected, even if just for a brief moment, will pop the light on and remain on. I'm not sure if it even shuts down the airbag system when it picks a fault so in even of an accident you'll get nothing - hazards of working with explosives I guess. I could be wrong on that though.
Sadly, resetting these systems isn't straight forward either. You'll need some software that can connect to the ECU and you'll need either a modified diagnostic cable or an adapter cable.
Most people use these OBD key / KKL (vag-com style audi) cables for diagnostics but they only talk to the engine which is on the K-line pin 7 of the connector. So you're only talking to the engine ECU, nothing else.
Annoyingly, the airbag ECU isn't on pin 7, it's on pin 3. So you either need to modify a cable taking the wire from pin 7 and connecting to pin 3, or get an adapter cable that goes from pin 7 to pin 3. Some people get clever and fit a switch so you can flick between the two.
The software you'll need to do it - most people use
MultiECUScan, formerly FiatECUScan. You should be able to access the Stilo airbag system using the free version of the software, if you can't then you're going to need to acquire the full version (around £40 I think).
Only other option is shout out your location and see if you can find a helpful person who has the cable(s) and software who doesn't mind resetting for you or paying a garage for the pleasure.
Good luck.