I'm also married to a Southern European and my fil also built us a holiday apartment. My understanding from this whole experience and my marriage in general is that it is very important to the older generation that property stays in the family. The best thing you can do for your child is to have them inherit some land or property.
We were also expected to invest in the apartment once it was built. I put my foot down and said no because it was crystal clear that as long as my pils are alive, that apartment will never belong to me. I have a sneaky feeling that my fil will put it in his will that my dd will inherit it, to prevent us ever selling it if possible.
I always refer to it as my fils/dhs apartment when in pils presence. With friends and colleagues I talk about our apartment, because in our culture it seems strange to refer to a spouses property as just theirs. But if my pils are clear on the fact it isn't really mine too, they can stop harassing us for money.
I remember having a big row with mil when I was refusing that we buy the fittings and furniture. She asked me why I wasn't grateful to be given an apartment like this. Well because it isn't mine, is it mil. I never asked for it to be built. I didn't have any say in how it was built. I didn't have any legal rights over the property. So thank you for letting me stay here once a year, but it also means we are tied to visiting you once a year for the rest of our lives, so we all have something to gain from the situation.
Your biggest mistake was investing your own money in this. You should have a conversation with your wife, and see who she thinks owns the house. If she agrees it's hers/pils then you should question the money you put in. If she agrees it's also yours, but pil doesn't like to admit it, then maybe just swallow your pride to keep the peace, and secretly enjoy the knowledge it really is yours and you own a property in Italy, which most British families don't.