I think that the weather in September is similar to the weather in June so in May it should be cooler, I'm sure someone will come along and correct me if I'm wrong. The only problem is that the weather seems to have gone weird in the last few years eg they had snow in april in Içmeler in april 2010. Have a look at
viewtopic.php?f=30&t=10170so the weather is not always as you expect it.
Our grandaughters were with us in July the eldest aged two and the youngest aged five months. The eldest loves the heat and for the youngest her mother had one of those folding UV tents which they can also sleep in at night as it has a mattress, as well as using it during the day as it has the protection from the sun. It was the third visit for the two year old whose first visit was at the age of five weeks and provided that she has the naps before becoming overtired she's fine. We spend a few hours indoors though to keep her out of the sun in the middle of the day.
It depends I suppose on the ages of the children. We always have air conditioning but you probably won't need it in May.