They have to record the songs ahead otherwise the sound would be a mess. Also, with all the different camera angles, and musical episodes always have like weird stuff happens, especially if there is dancing of some sort, the song would be choppy and not work at all. It is like when they do a music video. They record the song and then they have the singer act like they are singing for the music video.
They regularly have to rerecord dialogue on Ouat because the sound doesn’t get picked up properly on location. You can actually hear it during the show. Sometimes one line of dialogue will sound like it is outside in a lot of space and then the next line will sound like an actor in a box because that is what it is. The echo changes. There is just no way they could get proper sound quality for songs on location.
Honestly, the more we hear about this episode. The more horrifying it seems. I think i am just going to laugh through the entire episode.