I will give you my pov as someone who just cruised the Med. I am a long way from having a one year old, but, the process of cruising the Med is so different from say, a Caribbean cruise I will just offer my opinion, and, it is simply that, my opinion.
To take a Med excursion, means very long days. We had an excursion that lasted 10 hrs. Long bus rides,as long as 2 hours on the bus to get to our ports as well. Car seats and a carriage would be needed
The ports we visited had cobblestone and winding streets, many have steep hills, the simple fact of pushing a carriage up and down all these elevations, has to be tiring to say the least. Rest rooms to change a baby could be challenging as well.
Doing private tours in your case, if this is doable, would be the best bet, for convenience sake.
Coming back after these long days then dinner, might make room service look good.
I hate to be a Debby downer, but, taking one so young on these port intensive long days, sounds so tough vs, a more relaxing cruise for all three of you in the Caribbean, when, if you feel you need to get back to the ship, its easy... Maybe the Med,when the child is older?