Refrigerating medication while traveling in Italy and France

You can look for accommodations that have mini-fridges. Or you can stay in apartments that have refrigerators. And ice can always be acquired if absolutely necessary.
I'm diabetic and my medicines are best refrigerated. However insulin doesn't absolutely have to be refrigerated for a couple of weeks' travel. Check with your pharmacist to verify whether your meds must absolutely be kept cold.

More expensive hotels often have an in-room fridge/paid minibar. Cheaper places won't have that, but are likely to be open to refrigerating medicines (especially if you only need to collect them back once per day). Ice is not always handy to find at local mini-marts.

My parents are both on weekly injections of Humira which needs to be refridgerated. So when they/we go for a long trip, it's a pretty sizeable amount. They carry a cooler (not a hard one) as their personal item for a carry on. They use the packs that come with the Humira to keep them cool and toss right before security. That keeps them cold enough to last the flight. Every hotel they have stayed at have been willing to store the medication or to provide ice to add to the cooler. I don't think it will be a problem for you.

Ask at the hotel, I have never had a problem getting my meds taken care of. They may put them in a bag and in the fridge in the office or breakfast room, or restaurant if they have one. It really isn't a big deal for them. Then just ask when you need them. Ice is another matter. Depending on how long you are away, you might need to refill an ice bag more than once. Bars will do it but will look funny at you until you explain. Airlines will not let you put the meds in their fridge but will gladly fill your bag with ice. Refill upon landing!