A historicist architect can work off drawings without architectural plans. Perhaps more research is needed for the interior, but text depictions can suffice or a modern interior in the worst case scenario. The question isn't whether this is the best opportunity to rebuild a heritage structure but rather whether it's reasonable from a heritage preservation and historical commemoration standpoint.
Given the general ignorance about the site and its involvement in the War of 1812, it would seem to make sense to do so with the goal of preserving, commemorating, and promoting knowledge of our history and heritage. The modest nature of the building makes me doubt it would be particularly expensive.