The IATA guidelines for Lithium batteries for passenger luggage are summarised here.
CR2032 batteries are Lithium-methal batteries containing less than 2 g of lithium each (about 0.1 g, actually). As such, you are allowed to transport in your carry-on luggage:
- up to 15 devices with such a battery
- up to 20 spare batteries
Devices with a such a battery (but not spare batteries) are also supposedly allowed in the hold (as checked luggage), but it's unclear to me what the maximum amount is in that case (I would be surprised if there weren't any limits). At the very least, if carried as checked luggage:
• measures must be taken to protect the device from damage and to prevent unintentional activation;
• the device must be completely switched off (not in sleep or hibernation mode).
Note that individual airlines may have rules different from the IATA guidelines. I recommend you check with your airline for details. The Easyjet dangerous goods list seems to match the IATA guidelines, without mentioning on limit on the number of such devices or batteries.
Shipping lithium or lithium-ion batteries is also often an issue, with lots of stickers, warning and special procedures. They are often restricted to ground shipping. As such, I would not recommend you shipping them yourself.
My personal recommandation would be to carry the allowed amount, and order a bunch from Amazon IT and have them shipped to your hotel.
An alternative could be to use the equivalent rechargeable Lithium-ion batteries (LR2032), but be aware that their voltage/current/capacity may be different from non-rechargeable batteries, and they seem to be quite difficult to find.