Benjamin,
Margaret, and
Everett sound fine together to me.
Ben,
Meg and
Everett maybe don't go together so well, but I don't think that matters much. And if there are quite a few little Everetts around I doubt anyone from great-uncle's family is going to mind that you like the name.
Would you shorten it at all? I've never met an
Everett, so I've no idea if people do shorten it, or what to.