Is it permissible for the ruqyah practitioner to recite permissible verses of poetry secretly to the patient without their knowledge, while the patient believes he is reciting the Qur’an, for the purpose of discovering whether the case is suffering from delusion or a spiritual illness?

There is no harm in this action if there is an overriding benefit. The words of Shaykh al-Islam Ibn Taymiyyah indicate its permissibility for an excuse, citing the action of Abdullah ibn Rawahah (may Allah be pleased with him) when he swore to his wife and provided her with proof of his innocence from ritual impurity by reciting poetry that he made her believe was Qur’an. He mentioned this to the Prophet (peace and blessings of Allah be upon him), and he laughed. This action, if a man were to do it without an excuse, would be Haram by consensus.