1) Purpose: to illustrate the efforts of my young children in a non-professional context, not to showcase the audio portion of the recording.

2) Nature: audio + video, with explicit permission to record from the show executive/directors (it was a rehearsal, professionals recorded the actual show) and focus on the visual and physical nature of the unique performance

3) amount: abridged songs, amateurishly cut and remixed

4) effect on potential market: There is no way this is a substitute for any of the original copyrighted audio, and I communicate this intent to youtube by requesting it remain unlisted.

It probably does pass the four factor fair use test, but fighting with the youtube contentid machine is futile, and their system doesn't care about fair use.

- the purpose and character of the use, including whether such use is of a commercial nature or is for nonprofit educational purposes;

Which is non-commercial.

- the effect of the use upon the potential market for or value of the copyrighted work.

Which is to none.

There's also a question of if a video containing a bad transfer of a piece of music along with children dancing is "transformative." See Lenz v. Universal Music Corp, where Universal Music got destroyed all the way up to SCOTUS for ignoring Fair Usage when taking down a video of a dancing baby.

Your whole argument seems to be based on some fictional contractual agreement between the dance studio and the music's owners. That isn't relevant to the discussion at all, a parent recorded and distributed, not the dance studio themselves for commercial purposes.