OK so looking into this I see the car is pretty rare in that it has no passenger airbag, most likely because of being a very very early model.
the airbag switch issue as you say is irrelevant and isn't going to stop you connecting to the airbag ecu anyway
so the issue here is either
1. the airbag ecu which you have changed or
2. the cable you are using.
can you answer a couple of questions 1. have you always had the airbag light since you've owned the car?
have you tried another cable? if the switch isn't working properly or the cable itself isn't wired up properly you are never going to fix the fault.
I have a theory on this but you probably won't like it, but i need you to answer the questions first.
The original cable for the switch is very tight on length and sits above the glove box. Overfill the glove box and slam it and the glove box pulls the cable, pulls the wire off the switch triggering the airbag light.
You need to fix the problem by reconnecting the switch or even better fitting the replacement loom then reset the airbag light.
You must be new here? the switch is not very tight and does not sit above the glove box you can stick your hand up above the glove box if you like the only wire up there is the passenger airbag wiring and the car in this thread doesn't have even that.
the wiring is routed along the back of the dash at the very bottom where the glove box hinge mounts then up the side to the switch, it is under no tension.
the issue is entirely electrical and is the result of fiats poor wiring loom design often using under rated cabling, this causes a lot of reverse voltages through modules and puts a lot of noise on the earth cables. yes i know shutting the glove box (full or empty) can make the light come on however this i believe gives the whole electrical system in that area a bit of a shock with vibration just enough to tip the dodgy electrics already teetering, over the edge. it certainly doesn't pull any wiring or knock any wires off.
the whole reason I made the videos that cover this problem it because of all the rumours around what causes the problem and how to fix it without blowing your head off with an airbag.