somebody else singing a song is usually fine considering the amount of covers on YouTube
Just because it's on YouTube doesn't mean it's safe to do. YouTube has a whole range of agreements with different record companies so that in many cases copyrighted music will be used, but YouTube's contentID (I think that's what it's called?) system IDs it and reroutes the ad revenue to the "appropriate" record company.
IANAL, but I do know that compositions can be and do get copyrighted - see the avril Lavigne vs Rubinos case, where the allegation was that Avril Lavigne's song "Girlfriend" sounded too similar to one of the Rubino's songs. The case was, unfortunately for us, settled behind closed doors, but it goes to show how far copyright law can go. Which, TBH, is disappointing, because on the whole, I like the idea, but well, copyright ruins a number of things.
As to the state of Peertube, I've not run into any problems. I've mostly watched videos that The Linux experiment uploads himself to Peertube, but that has been pretty flawless recently. And especially in a case like this, which could potentially generate a significant amount of traffic, I don't see why it wouldn't work from a technical standpoint - more peers generally means those kind of P2P systems work better.
What I do worry about with Peertube is the whole copyright situation (again, copyright ruins everything). Since you're reuploading what you're watching to others, if copyrighted content were to be uploaded (which can happen very quickly in a music contest), it could potentially be open season for copyright trolls.
I don't want to be a downer, I honestly hate the way copyright ruins this stuff. But I think it's important to keep in the back of your mind in order to avoid it turning into more headaches than fun.