I’ve seen a question stating that you can listen to music for the purpose of a Mitzva - does that include a Bar mitzva? As a Bar mitzva date is predetermined at birth and is the main Mitzva for a lot of families, it would seem yes.
The music that is allowed according to some authorities is that which will lead towards a stronger bond with Hashem and will enhance the love and fear of G-d. The music in a Bar Mitzvah Party is usually for joy and dancing, which should not take place during the days of the Omer counting (there are different customs about which days of the Omer period). (See Igrot Moshe OC 1, 166) If the party is on the day the child turns thirteen in which case it qualifies a Seudat Mitzvah, and the music is of the nature described above one can follow a lenient approach and allow the music. (See Shulchan Aruch OC 493 and commentators.)