I think it Has been answered if I understood correctly.
no, the patronymic doesn't change into middle name, in other countries documents it just doesn't appear, and I think if the child will have double nationality they will register the patronymic in the Russian documents, and if they want to give the child a middle name can be whichever they want.
My mother and I don't fit you in your American related example, but in any case No patronymic figures as a middle name in our Italian documents (she has double citizenship as well and she 100% has the patronymic). Outside Russia and Russian speaking country it just doesn't figure.
I think would sound really weird a Ivanovna or Lvovna as middle name in an English speaking country but hey, people call their children apple so I guess everything is possible.