The word Caprice is really the French form of the Italian Capriccio. I think Italians should use the French pronunciation, because they are using a French form, but perhaps they don't, I don't know.
This word has a rather surprising origin. It literally means 'head (like a) hedgehog', which is to say, with the hairs standing upright (from fright). This is the original meaning: a horror. (
Oxf.Engl.Dict.)