BYM: If someone asked me, I would say that it’s an Issur D’Oraisa of Darchei HaEmori.
But it is found in 1 (exactly 1) very recent T’shuvah Sefer and his reason for the Minhag is that… SheKein HaNashim Makpidim Al Zeh. (In an adjacent T’shuvah he mentions that he never let his children see their reflection before they grew teeth because otherwise their teeth would never grow in.)