A novel like 'The War of 1812: A Short History' can be considered. It might be more of a historical account presented in a novel - like style. It details the causes of the war, such as the trade restrictions and the impressment of American sailors by the British. It also describes the major battles like the Battle of New Orleans. Reading this can give you a good understanding of the overall picture of the War of 1812 in a fictionalized but still accurate way.
Yes, 'James Madison and the Struggle for the Bill of Rights' is a great historical novel related to the War of 1812. It not only covers the war itself but also the political situation around that time. It gives a comprehensive view of how the war was related to the domestic political struggles in the United States. There are also some novels that tell the story from the perspective of ordinary soldiers, which can provide a more personal view of the war.