The Brazilian Imperial Family is Catholic, as are many of the Germanic houses.
If Harry wanted to find a Princess to marry he would have to either look outside of Europe or look at deposed royal houses. I believe in the Middle Eastern royal houses there are a number of princesses about Harry's age, although I'm not sure if they're single or not. There are also many princesses (or grand duchesses) from deposed royal houses who are potentially pressured into marrying equally because of the continuation of the old house laws.
However, while I believe that there could be an advantage for Harry marrying someone like, say, Madeleine of Sweden, I'm not sure that advantage extends to a Princess from a non-ruling house. Madeleine would have been exposed to a life more similar to what Harry's experienced than a non-reigning royal or a non-royal. She has an understanding of what is expected of a royal that someone like Kate wouldn't have had going into the marriage. As such, there theoretically should be less of a learning curve and she should have an easier time adjusting to the role - although the question of how much time do they spend in Britain and how much time in Sweden, when are they representing Britain/the Commonwealth and when are they representing Sweden, etc, all comes up.
Harry or William could have both done well by marrying the daughter or granddaughter of a reigning monarch, but that opportunity has kind of passed.