8th Amendment?
I think that's the ticket. If a state handed down the death penalty for something, say theft, I would expect the 8th Amendment to be cited when it was overturned.
Why not set forth a specific set of circumstances where tge death penalty applies, if convicted? And if the conditions were met, then the ONLY penalty is death, none of this "life without parole" stuff. That way no one group is unfairly subject.