Carol would have had no reason to confess to Tyrese after Lizzie was dead, and risk him shooting her.
I think in that very moment, when she slid the gun to Tyrese, she wanted to die. She had just killed a child - a child that she fought like hell to protect because what is the point of living if there are no people after you to carry on.
I think this was her very most lowest point...rock bottom - and she had given up.
On the other hand - her will to live is permanant and therefore - her use of the confession to push Tyrese over the edge.
Also, i think that her guilt with helping to dispose the bodies was there.
Plus why would Carol leave Lizzie at the prison without telling anyone if she had killed Karen and David?
The same reason why she accepted the banishment. To protect the children. Even though Lizze did this horrible thing - she was still a child and Carol couldn't accept that Lizzie was 50 cards short of a full deck.
And why would Carol have accepted being banished if she didn't do it?
Denial is not just a big river in egypt and I really dont think that Carol wanted to believe that Lizze actually enjoyed or like killing people and wanting ot be with walkers.
If Carol said "but i didnt do it" then it would put Rick into the position of "so you are telling me a child killed them, you are supposed to expect me to believe that??"
Rick would have believed it and it would have then put Lizzie's life on the line.
It was a lose lose here