A Brit Milah feast is a Se’udat Mitzvah (a mandatory halachic feast), as explained in the Talmud and Shulchan Aruch. The Halachic authorities dispute whether singing and dancing can be permitted at a Se’udat Mitzvah up until Rosh Chodesh Av. The author of Shiurei Knesset HaGedolah (Siman 551) permitted bringing musicians, dancing, and singing, and Maran the Chida (Chaim Sha’al, Vol. 1, Siman 21) and the Mo’ed LeChol Chai (Siman 338) ruled similarly. The eminent Torah genius Rabbi Menashe Klein, in his book Mishneh Halachot (Vol. 6, Siman 109), cites Rav Moshe Feinstein stating that the same law applies to a Sheva Brachot feast. However, the Mishneh Halachot (ibid.) notes that some allowed dancing and circle-dancing without musical instruments. Others disagreed and were very stringent on this matter (Responsa Shraga HaMeir, Vol. 2, Siman 13). Furthermore, it is commonly cited in the name of the Steipler, zt”l, that it is permissible to sing using the mouth alone (acapella).

In practice, the author of Kaf HaChaim determined that for one who is lenient, and in places where they are lenient, there is authority to rely upon. Therefore, it is certainly possible to be lenient both in hiring a DJ and regarding singing and dancing. We must qualify that this rule applies to Ashkenazim until Rosh Chodesh [Av] and to Sephardim until the week in which Tisha B’Av falls, after which musical instruments are forbidden even at a Se’udat Mitzvah. However, one must know that during these weeks our Sages instructed us that the House of our God is worthy of being mourned over; thus, in whatever way you can, and if you are able to be stringent, may a blessing come upon you.

All of the above applies equally to both a Brit Milah and a Pidyon HaBen (Redemption of the Firstborn).