So you got permission not only from her family but from the government to marry this Qatari lady? This is a very rare occurrence. I am assuming you are muslim as she would not be allowed to marry a non-muslim man. In that case, I would assume you are governed by the Shari'a legal system in place making rulings over child custody. The man always wins out UNLESS her family is very powerful and tries to exert some influence.

If your marriage is a good one - why are you worrying about losing custody of your children? Wouldn't your wife want your family to stay together.