Is there a mental defense they could use to try to excuse/explain his cover-up behavior after the crime? Something like PTSD caused by seeing his girls killed?
I really think this is more likely to be about a possible competency exam. And as KathrynL discussed- a heat of passion type mitigator. That makes sense because his state of mind, emotional issues, mental health, would or could go to both.
Besides something like a psychosis so grave he didn't even know what he was doing, I can't think of a diagnosis that could help him in an insanity defense. Or explain his post-crime behavior, really.
The insanity/diminished capacity defense typically means that due to any mental disease or defect, the person did not know what they were doing - so the one they teach us in law school is they think they're squeezing lemons but they're actually squeezing someone's neck - or they don't realize that what they're doing is wrong. So they're so incapacitated by insanity or brain damage of some sort that they think what they're doing is legal or otherwise wrong.
An example of that is Andrea Yates. At her re-trial it was found she did not understand that what she was doing was wrong. She knew it was illegal however she felt it wasn't wrong because she thought she was saving her kids from eternal hellfire. So she was found not guilty by reason of insanity.
She was incredibly psychotic.
CW showed zero signs of any illness right after the murders from what we can see from his behavior - what he did, said, how he acted, etc.
Yates didn't try to cover up the crimes. She was practically catatonic in her affect. She called 911 and immediately admitted what she had done, which is typical of the truly insane.
They usually lack the capacity to try to conceal what they did.
But if you just mean they may try to use something to simply try to show why his post-crime behavior doesn't exhibit consciousness of guilt, that could be.
I mean they had experts on to prove that casey Anthony's behavior in partying and getting tattoos didn't show she was a heartless murderer but that everyone "grieves differently".
Hell. I didn't know some parents grieve by dancing on a stripper pole.