To Live as a Monster
If he accepts his guilt and becomes sane he will become the Monster. His delusions are his brains way of removing this guilt and hiding the fact that he ignored his family, did not act on his wife's mental illness, allowed her to murder their children and then murdered her. By accepting what the doctors are telling him he will need to go on living in the knowledge that he allowed terrible things to happen and did terrible things. The doctors will be able to treat him without a lobotomy but he will be fully aware of the crimes he committed.
To die as a good man
In his delusion he is a good man. He truly believes that he is on the side of law and order and that he is doing right. If he chooses to continue to believe this he will be lobotomised and will no longer have relevant brain function to question what he did or did not do. The death is not literal as the other answer suggests. It is about the fact that if he continues to believe he is this good man he will be lobotomised and effectively "die" believing in this. He will not die in a literal sense until later, but he will do so in the belief he was a good man.
What is most exciting and cathartic about this decision is that this appears to be a moment of lucidity where he understands what has happened and is not in thrall of his delusions. He at this point of lucidity makes the decision that he can not live with his crimes and would prefer to lie to his doctor as he knows this will result in his lobotomy.
By making the statement to his doctor he makes him aware of this also and by virtue of this you could also argue that he is being a good man by electing to be punished as well as the belief in his delusion that he is good.