Not a lawyer. That said, I gave your video a quick look.
Red flags (to me)
Song seems to be overlayed on to the video. It didn't seem like it was playing in the background, because the sound did not change as the camera/microphone was moving and there was no wind/external noise as there was during some of the speaking parts. I agree it was probably the right call as changing sound levels and background really annoy people and would have seriously damaged the quality of the video, but with it being overlayed, it probably strengthens their claim. (it wasn't just background music, it was added in after the fact)
Advertisement at the end: This is probably the biggest issue. Again, it seems like you are advertising a dance studio. There is no "Special thanks to" with the dance studio. Just the studio name, logo, and contact information. Perception is reality. If a reasonable person could say that at least part of it was advertising, even if it wasn't, they can probably get you.
It was a great proposal. The video was shot well. The routine was great. I am not sure if you can get any kind of fair use, but see if you can get a license to use after the fact (aka, talk to them, apologize, explain things and ask if you can buy the rights use the song for use in that one video..especially if you change the end to a special thanks (removing logo, etc) to the dance studio and promise to not monetize the video yourself. You might even want to bring your fiance/wife
I wish you the best of luck with it. If nothing else, keep the video and you will have something great to show to family and friends down the road when you hit your 25th and 50th anniversary.