The conclusion is Fiat stop start system is plagued with problems.
My brand new 500 would stop and restart within seconds.
My panda has had problems since new for 2.5 years. The thing will turn off, but intermittently will not restart, forcing a rapid turn off and back on of the key.
Fiat are useless, and are not interested in trying to fix it.

I've come to believe that the only use for it is to lower my annual car tax bill, and I've given up trying to get it fixed. I suggest you do the same as the hours invested will not be worth it.

My start stop on a 2012 TA has been faultless, the only time it doesn't work is when the car has been left for a few weeks undriven and then it won't work until the battery has had time to recharge itself.

Fiat don't expect it to work every time. Bloke at the local dealership told me. Meh.