What an idiot, he's just turned down the most humane and painless way to go. You don't notice nitrogen suffocation, because your body ignores nitrogen in the air and determines you're suffocating by a build up in CO2. Instead, you pass out in blissful hypoxia.
I'm against the death penalty as a rule of thumb, but if you have to do it then it should only be done via nitrogen suffocation. Anything else is just a refelction of the vindictiveness of the people administering or pushing for the punishment - it doesn't achieve anything, it doesn't deter future crime, it's just you getting your own back and trying to say it's ok to harm others in this instance. If the goal is to remove them from society such that they don't harm or cost society anymore, then this should be done without the kind of harmful intent that the criminal themselves demonstrated.
Tbh though I imagine this is just the guy's lawyer trying to do anything he can to delay the execution. There's some small chance that the state could do something wrong during the hearings that leads to some benefit for the prisoner. However I can only imagine the regret the prisoner might feel as he's on the receiving ends of one of the other methods.