No ideas on specific songs yet, but Sweeney Todd is a hybrid show. It is a cross between an old-fashioned 19th century melodrama and an opera. Since both of those genres depend on characters that are larger-than-life and over-the-top, look for a song that can be done that way. Have a POV.
Rose's Turn (perhaps not specifically appropriate) is something like this. The character uses the song to work out a solution to a personal situation and the singer ends the song in a distinctly different place than where they started.
Sweeney after his 'Epiphany' is certifiable. That's a different place to be from sane at the start. Find a song that you can go down the path to lunacy, but a scary crazy which by which I mean one that isn't that far removed from everyday 'normal' behavior.
Treat it as an exercise. Have fun.
"If my life weren't funny, it would just be true. And that would be unacceptable."
--Carrie Fisher