May is not beach weather. In fact we have been in Italy several times when it was cold and rainy in May. It has never when we were there been warm enough to swim in a pool much less the ocean.
If it were me (when it was me) I would rent a place in a small Tuscan hill town and have a car and spend the time visiting other towns, gardens and monasteries in Tuscany. Here are a few examples from when we last did this although it was September not May. https://janettravels.wordpress.com/category/tuscany/ In earlier years when I was still working we always visited Europe and mostly Italy the last 3 weeks of May. We like traveling in cooler weather but it is definitely not beach weather; when we rented a place that had a pool it was not really usable that time of year.
There is no point visiting Venice for one day or Rome for two days. Not much point of that much time in Amalfi that time of year unless sitting and looking at the view is what you want to do. (not a bad occupation) When we did this last, we flew into Rome, spent 3 nights to see a few things we had missed on earlier trips, and then drove to Montepulciano for two weeks. On an earlier trip, we flew to Rome, picked up a car at the airport and drove 50 miles around Rome to the Villa Adriana in Tivoli, stayed at the hotel at the gates of this site, and spent 'jet lag day' wandering around the fabulous ruins in the sun. After a good nights sleep with drove north to a rental place near Lucignano for a week's stay.
If you want to see Venice and Rome and Florence in such a short time, then drop Amalfi -- it would be pretty tight even without that. We spent 5 nights in Venice in early May 35 years ago and the weather was perfect and it was lovely -- but May is always dicey -- sometimes pleasant and sunny but often rainy and cold.
Remember that travel days are complete shot and you lose a full day on the first day of travel i.e. day 1 is travel arriving on day two and the last day is all travel. It takes two nights to get one full day on the ground and every time you make a major city change you chew up most of that day. for 13 days total, I'd choose two locations at most and focus on maximizing time on the ground. Certainly no more than 3.