Is the action of the shaykh who allowed the wine drinker to finish his bottle in exchange for listening to his da‘wah correct, given the possibility of the drinker dying in that state?

An intoxicated person is prohibited from entering the mosque, whether they are a Muslim or a non-Muslim, even if merely passing through, and they are prohibited from residing in it. However, if their entry leads to a greater benefit than the harm of their entry, then the rules of Sharia (definition) dictate that they should not be prevented. If a caller [to Islam] permitted him to drink alcohol in the mosque, the matter requires verification of the circumstances of the incident. If it is proven that he permitted him to do so and was able to prevent him, then this could be interpreted as his intention to prioritize the greater of two benefits, which is inviting him to abandon alcohol and explaining its harms, especially if the man would not listen to him until after he had finished drinking the alcohol, and it was not possible to prevent him from it.