Is it permissible to prevent a non-Muslim man who is not in a state of ritual purity from praying in the mosque, especially since he rejects Islam and his condition has not changed despite the advice of one of the shaykhs to leave him be, hoping that Allah guides him?

First: It is permissible for a non-Muslim to enter a mosque if there is a legitimate benefit and if he does not defile it.

Second: It is not permissible to allow anyone to pray without ablution, whether Muslim or non-Muslim, as non-Muslims are addressed by the branches of Islamic law.

If a non-Muslim performs ablution and prays:

If he openly declares his disbelief, then it is apparent that he should be prevented from praying, due to the disrespect this entails towards the prayer and its disruption for the worshippers. Moreover, if he were to be judged as a Muslim (initially), he would then be considered an apostate if he openly declares his disbelief, and the punishment for an apostate is execution. If he does not openly declare his disbelief, it is possible that he is inwardly a Muslim.

This man should be invited to Islam. If he insists on his disbelief after the explanation, then he should be prevented from entering the mosque because there is no benefit in his frequenting it.