I think if you have the Anti-federalist elements win out, this could be the result (it's certainly a necessary precursor). Taking the Swiss model as closely as possible, we have 13 states, in perpetual alliance, with a figurehead ruler and consensus based government, each essentially independent and having complete control over her own domestic policy and the state militias being the only army. No Congress or Senate at all- national policy is determined by the heads of the states meeting together.
Now, this is obviously unworkable due to the much greater size of America compared to Switzerland, so some elements will need to be added. Potentially this would be a single house of representatives from the states, but which has control over precisely three areas: Foreign Policy, the Continental Army and international tariffs. The continental army exists to supplement the state militias, but those are the primary military force (note that a lot of this would go in Switzerland after Napoleon so might not last that long in the US either). The President is probably an entirely ceremonial figure with the role of Commander in Chief of the CA, but he may also preside as a neutral figure in the Senate.
This might be just about workable, but the US may well end up fragmenting under this government.