You are correct that burden of proof is left to the State not defense, however, it is not that simple. The jury decides regardless, THEN, insufficiency comes into play in a normal trial. A DP trial appeals are automatic.
As far as undiagnosed mental illness, well, a lot of us have issues, many more difficult than other people, but we still did not kill our child. (Not everyone with PPD kills their child even if they think of it, they stop themselves, and I don't know about any possible abortion, but KC was past time for PPD. In other words, the jury will not buy it. PPD afflicted most likely do not go out and party and find new or several boyfriends, they just flip out.) Our issues may have been more difficult, more compound, whatever. I am not saying KC did not have issues, mostly fighting with Mom for control and inability to accept responsibility and move out on her own, and I felt sorry for her a while, but I realized, it is worse, so much worse, for others.
I am wondering how a jury will view whatever is presented. DP is not going off the table, trial is going forward with DP now. AL is in unless AL chooses to go, it is going forward. The only thing KC can do is plea to get DP to go away. I wonder if the team will get through to her to spare her life as you have questioned. I don't know, in the state I live in it is rare for the person to plea and save themselves, even when they have given several different versions of confession.
Just hope any of that made sense.
P.S. I still think that if she is deemed death penalty Judge S will set aside. JMO.