Is it a sin if a Catholic Judge sentences someone who is proved guilty as per law to death?

The judge is not committing sin  in ensuring the rule of the law provided the law of the nation has formed on a sound moral conscience. In case, the law is formed on a doubtful or a bad moral conscience then the moral culpability of the judge is subject to his degree of cooperation with the state. In all other instance where the law is formed on a sound moral conscience the judge is free from sin while ordering a capital punishment. 

However it remains a valid question whether capital punishment is legitimate for any heinous crime. Catechism 2267 has been amended in Aug, 2018 saying that the death penalty is now “inadmissible” which was admissible until recently. It is perceived inadmissible due to a change in the ‘circumstances’ of the act of ‘executing death penalty’ which until recently necessitated, though only rarely, the said act. It proceeds from a more comprehensive understanding by the Church on the efficiency of penal systems of states. The Church perceives that the modernization of the penal systems has been empowered them to preserve common good without actually executing the guilty. 

The object and intention of death penalty remains same, only the circumstances changed, that too as per the understanding of the Church which is again limited and fallible in this regard. Hence, the amendment isn’t doctrinal. The change is largely based on Church’s understating on the efficiency of modern penal systems. The penal systems of third world countries aren’t as effective as developed countries. In certain situations, say when a dangerous terrorist is caught, poor countries are left with no option other than a death penalty to preserve the common good. 

It’s the duty of the government to improve their penal systems and modify the legislation to totally avoid the need of death penalty. Until then the judge is morally permissible to order death penalty to protect the rule of the law.