One of the darkest musicals I like to listen to is Murder Ballad. Listening to this show is like walking down a dark alley alone at midnight.
It's not a musical, but there's an opera called Doctor Atomic that's about the inventor of the Atom bomb, and his turmoil over creating the most destructive force ever made by humans.
It's hit or miss for some, but The Devil's Carnival (by the creators of Repo: The Genetic Opera) is supposed to be pretty good.
It's more of a romantic drama, but Sondheim's Passion is said to be his darkest work of theatre, and it deals with mental illness.
Nevermore: The Imaginary Life and Mysterious Death of Edgar Allan Poe. It was a hit in Canada and ran Off-Broadway a year or two ago and is available through Playscripts Inc.
OMG Ride the Cyclone, that could DEF qualify. I don't know a whole lot about it, but what I do know, it seems eerie as fffucccckkkkk....
Also this is such a specific thing you're looking for, and honestly there's not a great amount of shows that check off all those points. I would say the most "mental illness" themed show is certainly N2N, that's literally what it's about obviously, but Carrie, Sweeney Todd, and maybe a couple others, qualify as the "bloody" ones, though "mental illness" isn't the foremost theme of the show. Frankly, anyone raised like Carrie (and we all know they exist...) would certainly not come out of that with any degree of good mental health...OI!