How can one object to determinism when the Preserved Tablet supports it and negates human choice in belief and disbelief?

This is an old misconception, and Sheikh Al-Uthaymeen responded to it, clarifying that the determinists' (Jabrīyah's) reliance on verses such as: "Allah is the Creator of all things," "And Allah created you and that which you do," "And you threw not when you did throw, but it was Allah Who threw," and "Those who associated partners with Allah will say: 'If Allah had willed, we would not have associated partners'" is invalid. The first verse is contradicted by texts that affirm the servant's will. The second verse is an argument against them because it attributes the action to them, and Allah's creation of the action is due to His creation of the will and ability. The third verse affirms the throwing to the Prophet (peace be upon him) and attributes the delivery to Allah. The fourth verse invalidates the polytheists' argument.

He then demonstrated the invalidity of the doctrine of determinism (Jabr) by means of the Qur'an, the Sunnah, reason, sensory experience, and the consensus of the predecessors (Salaf). The Qur'an affirms for the servant will, speech, and action. The Sunnah confirms freedom of choice. The consensus of the predecessors did not transmit from any of them the قول (saying/doctrine) of determinism. Reason dictates that determinism would render punishment unjust and reward futile, and would invalidate the argument for sending messengers. As for sensory experience, it distinguishes between voluntary and involuntary actions.