Technically, a lightweight upper structure would be feasible and were actually typical on 19th century ships, the American Charles W. Morgan for instance was built flush decked, then had deckhouses added, fore and aft, around mid century.
But I doubt that they would be used on a man of war in the late 18th century, especially a Commodore's vessel. Social status was simply too important. It would be difficult for a Commodore or Captain to accept the indignity of living in a shack on deck. The symbols of authority and rank probably would have overruled technical practicality. For instance, when the Royal Navy's 20 gun ships were elevated to ‘Post Captain' status during the middle of the 18th century, beakhead bulkheads and quarter galleries were installed, big ship features on little ships, but symbolic of post rank. These features were regarded as socially necessary, even though they degraded the sailing qualities of what were already mediocre ships.
It's hard for to comprehend such social anachronisms, but they persisted into the 20th century. For instance, up through World War II British destroyers still housed the officers, aft, and non-coms, forward, per Naval tradition, even though they had to reverse their stations on ship to get to their duty posts! It's hard to say how many sailors and officers were washed overboard in storms or how battle efficiency suffered from the resultant delay in ‘beating to quarters' in honor of this tradition.
My belief is that the aft end of the United States would have resembled the stern of a 64 or 74 gun ship (vessels unavailable to the fledgling US Navy), befitting the status of a Commodore.
Of course we may never know. Until primary evidence comes to light we are stuck with conjecture.