My husband's entire extended family are known by their second names, it's a common practice in many areas,
As most of them have the same first names, and same surnames, the tradition is that they go by their second names, or you would have a lot confusion.
Just use the first initial, second name and surname.
so David Henry Smith, becomes D.Henry Smith,
Then no legal change is necessary, my husbands bank accounts , have the equivalent to Mr.D.Henry Smith, everyone knows him as " Henry", schools always ask what a child's chosen name is, when they start.
The point of having second, third names is so you can use them.