Well tyre choice is often very emotive and difficult to nail as is to what is best, better, etc.
What I can say is that on a 2005 'B' the originial Pirelli P6000 tyres were absolutely great.
As P6000s are no longer available we have Pirelli P1 Cinturato 195/55 R15 85V tyres on the front and P6000s on the rear.
No problems or issues, wet or dry and we (me and wife) have both done over 6000 race track miles and we don't hang about on public roads when it is safe to open the envelope within legal limits.
What is very difficult to ascertain / judge is if a P6000 / Toyo Proxes / P1 or any other tyre is better or worse than an other unless you can do back to back comparisons.
For us P6000s & P1s appear to be very similar, balanced and no surprises. We have had major spin surprises on our Strada Abarth with Goodyear NCT tyres compared to Pirelli P600s and other brands.
The point I'm trying to make is that any given car and any given tyre is going to behave totally different to another car/tyre combinations.
What we can see here for the Barchetta is that:
Pirelli P6000's
Pirelli P1's
Toyo PROXES T1-R
get good votes/support.
If you search the forums in detail other tyres may also get the thumbs up.
Just be careful to check like for like. e.g. owner A said that Tyre A was inferior to Tyre B. This does not mean that B is the best tyre. Some owners my even prefer A to B. Meanwhile tyre C could be the best tyre in some peoples view but unless they have driven/evaluate A,B,D,E etc. then whilst they are perfectly happy with C it could be the worst or best tyre you have ever had on your car.
Some tyres will shine through with "worst", "lethal", and other completely daming reports. These are the only tyres you should avoid. For you one of these tyres could be the greatest tyre ever, but it is very unlikely you will hit on that tyre.
My advice. Do as much research as you can. Balance price and availabilty and brand quality together with customer and more official industry tests/reports as you can.
From my point of view, generally on most Fiats Pirelli tyres seem to be on balance a safe, but not necessarily the best, option. On our Fiats (over 40 years) we have had Pirelli, Goodyear, Bridgestone, Michelin, Toyo, Yokahama, Avon, Dunlop, Esso, Hankook and Kumho tyres (that I can readily recall). Non have disappointed except for the Goodyear NCT1 tyres on our Strada Abarth. However the same Goodyear OEM NCT1 fits on our Stilo Abarth did not present any major issues.
Hope you find a tyre that meets yor requirements and expectations. Don't forget to honestly post your findings back. If you end up buying a crap tyre then let us all know. Their is no shame in getting a choice/purchase wrong. We all get it wrong sometimes. The important thing is to let people know especially when you have got it wrong. Info on failures is worth 10 times more than info on successes.
