It seems to me from the description of the steering swinging out with the front opening door, that you are in fact describing not a FIAT but a BMW ISETTA.
This was an Italian design but German built car with a 300 cc air cooled single cylinder.       I owned 2 years ago in the UK.       As it had 3 wheels you had to pay only motorcycle road tax.      It was practical transportation but I nearly rolled it several times.

I have also owned FIATs and have enjoyed them .      The car that I have owned that was the most fun to drive was in fact a FIAT 128 coupe.     Drove just like a go kart!

Memories from the past!  

Steve Cotsford