Best month for southern Italy tour

Our preference for travel is mid-Sep to mid-Oct. We generally find it dryer and cooler than summer and warmer than May.

I have never done the tour, both months are pretty good, but I think September would be the better option.

Generally September is hotter than May, especially early September, but temperatures at the end of May will likely be similar to temperatures at the end of September.
September is often wetter because Italy gets heavy downpours at the end of Summer.
In May the sunset comes later therefore you will have longer days in May, especially at the end of May.
Both can be good choices, but the weather often changes drastically between the beginning and the end of both months. Early May is generally cool, late May is likely warm to hot. Early September can be unbearably hot compared to end of September.

Below are weather averages for Naples. Bari, on the Adriatic coast, will be similar.
http://www.holiday-weather.com/naples/averages/

As Roberto says, September is usually warmer (and can be hot) compared to May.
If swimming in the sea is on the agenda, definitely choose September.