Let me introduce myself enough to say that I am an ultra-orthodox (or preferably hareidi) rabbi, a mesader gittin, shochet, mohel, sopher.
Shulcha Aruch Yoreh Deah 268:3-4 deals with this subject. The Shulchan Aruch begins by saying that all of the requirements of a convert must be done in front of a beis din of 3 including informing the convert of the mitzvos, the circumcision and the immersion. He then brings the opinion that the milah and tevila could be not in front of the beis din and concludes with the opinion of the Rambam and the Riph that if they weren't in front of beis din, the conversion is not valid. In paragraph 4, he continues "Since the immersion of a convert requires beis din we don't immerse the convert on Shabbes or Yom Tov or at night." So paragraph 4 assumes the opinion of the Rambam and Riph that the tevila and mila must be before the beis din.