The Shulchan Aruch (183:7) says that one should Bentch along *quietly* with the Mevarech. The Mishna Brura (sk 27, 28) explains that one should Davka say it quietly with him, and not out loud, everyone by himself, like some people do. (a) Is the problem saying it out loud, or everyone by himself even quietly? or maybe both? (b) (this depends on the answer to part a) Is there a problem with singing bentching out loud with one’s children, even if it is along with the Mevarech?
The Halacha you referred to applies only when there is Zimun and one person is leading the Benching, and only to the first Bracha till "Hazan Et Hakol" (1). a) The main point is that everyone says the first Bracha together with the leader ("Mevarech") (2). If everyone at the table joins in, there's no problem saying it together even out loud. The problem is when everyone benches alone at his own pace (3). b) Therefore, there's no problem singing with the children, as long as it's together with the Mevarech, Sources: (1) Mishna Brura, 183; 28 (2) ibid. 183; 27 - "ונרוממה שמו יחדיו" - together. (3) See Chayei Adam 48;1 - "מה שהעולם נוהגין שכל אחד ואחד מברך לעצמו - זה לא נקרא כלל ברכת הזימון".