Should the tenant register the house and pay the bills despite the owner's refusal, given that he insists he is responsible before the government and God?

If the state has exempted him only from the costs of his personal consumption of water and electricity, then it is not permissible for him to prevent you from paying for your consumption costs, or for him to benefit from their price, because that is a betrayal of trust and a violation of the condition. Nor is it permissible for you to obey him in not paying.