I actually prefer the French pronunciation over the Finnish one. The latter sounds exactly like Henry. So I prefer Henri and Anri. It's handsome and gentle.
Personally, I think that Henri could be given as a child's nickname when the boy is 5, but after that, he should stick with the Henry spelling. I mean, it's cute all right, it just wouldn't age the best.
Henri is not an alternate spelling, it is the standard spelling in French and Finnish. The pronunciation is also far different in French. The English form of a name is not the standard for everyone. Henri is perfectly normal in different languages, and is not childish. I think it's time you consider there are languages and cultural standards different from your own.
I like this way better than Henry because I think the French pronunciation sounds richer in some way. It was my late grandfather's name (he wasn't French or anything, it's just sort of a tradition in my family to have a French name) and I might use it as a middle name for a son later.