Struwwelpeter, by Heinrich Hoffmann. He doesn't get thrown out with the rubbish, tho; people merely say "Pfui" to him. Most of the naughty people in the book do not end up dead, but the girl who plays with matches ends up as a pile of ashes, and a hunter who falls asleep is shot by a rabbit who steals his gun.

The introduction to _Through the Looking Glass_ ends

Come, harken then, ere voice of dread,
    With bitter tidings laden,
Shall summon to unwelcome bed
    A melancholy maiden!
We are but older children, dear,
Who fret to find out bedtime near.

-- and the prose of the Alice books is full of jokes about death.

Mother Goose is full of death, mostly violent & grotesque. Who killed Cock Robin? Most of the versions of Froggy Went a-Courting end in death. Likewise

The little dog he has burnt his tail,
And he must be hanged tomorrow.

*

And when your back begins to smart,
It's like a penknife in your heart;
And when your heart begins to bleed,
You're dead, and dead, and dead, indeed.

*

On looking up, on looking down,
She saw a dead man on the ground;
And from his nose unto his chin,
The worms crawled out, the worms crawled in.

Then she unto the parson said,
Shall I be so when I am dead:
O yes! O yes! The parson said,
You will be so when you are dead.

*

Fee, fie, foh, fum....

*

Ho! Master Teague, what is your story?
I went to the wood and killed a tory....

*

There was a little one-eyed gunner
Who kill'd all the birds that died last summer.

And, best of all:

Baby, baby, naughty baby,
Hush, you squalling thing, I say.
Peace this moment, peace, or maybe
Bonaparte will pass this way.

Baby, baby, he's a giant,
Tall and black as Rouen steeple,
And he breakfasts, dines, rely on't,
Every day on naughty people.

Baby, baby, if he hears you,
As he gallops past the house,
Limb from limb he'll surely tear you,
Just as pussy tears a mouse.

And he'll beat you, beat you, beat you,
And he'll beat you into pap,
And he'll eat you, eat you, eat you,
Every morsel snap, snap, snap.