Have you ever been to Europe?
Cruising the Med in the summer will be very hot! I would go in June.
Even in early Sept. it was over 100 degrees in Rome and the Greek Islands.
On a ten day and eleven day cruise we had only two sea days. Not much time to be at the pool if you get off at every port of call.

The suggestion of DYI in the ports with a five year old is good advise.
Most private tours and ship tours are all day, with lots of time in a van or bus to get to the local venues.
The ship does not really matter, because the whole point of a European cruise is the ports of call.
would you leave your child in the kids club while you get off the ship? Or will you be traveling with other family members?

Sorry for all the questions. but I have grandchildren and my five year old was overwhelmed by Disney World ( needed down time, nap )compared to the older grandchildren.