Seems that you're finding the bookmark correctly, but then copying it wrong. You're selecting (highlighting) the contents of the bookmark, but then actually copying the contents of a
different range (when you go
objWord.Range(ref oMissing, ref oMissing)
you're basically identifying the entire document body, because you didn't specify the start and end positions).
I think the correct code would be:
objWord.Bookmarks.get_Item(ref oName).Range.Copy();
wordApp.ActiveWindow.Selection.Paste();