Is it permissible to interrupt the Dhuhr prayer after the second Rak'ah with the intention of converting it to the Fajr prayer for someone who remembers not having prayed Fajr?

According to the three Imams—Abu Hanifa, Malik, and Ahmad—it is obligatory to observe the order when making up missed prayers. Their evidence is the action of the Prophet, peace and blessings be upon him, on the Day of the Trench. However, the obligation of observing the order is waived due to forgetfulness, based on the Prophet's saying, peace and blessings be upon him: "Indeed, Allah has pardoned for my Ummah their mistakes, their forgetfulness, and what they are forced to do."

If a worshiper forgets a missed prayer: 1. If they remember it before the current prayer: They should start with the missed prayer, then perform the current prayer. 2. If they remember it after performing the current prayer: Their current prayer is valid, and they only need to perform the missed prayer. 3. If they remember it during the current prayer: They should complete the current prayer (it will be considered a supererogatory prayer), then perform the missed prayer, and then repeat the current prayer to maintain the order. Alternatively, they may complete the current prayer and then perform the missed prayer without repeating the current prayer; this is the view of Imam Al-Shafi'i and the preferred opinion of Ibn Uthaymeen.

As for abandoning prayer due to illness, this is not permissible. It is obligatory to perform it at its due time to the best of one's ability.