With Jefferson as President, I don't see the US as likely to declare any kind of war.
Note that Great Britain may not have been as economically drained in 1804 but are certainly disheartened, having been beaten in Europe for over a decade by this point.
Also note that this point may be the best shot of an American-French alliance as Jefferson was in power (an ardent Frenchophile) and Europe was at peace. By this point, many other prominent American Polits were tired of the violent Revolution (again, Jefferson not so much) and probably looked forward to the "stability" that the newly self-crowned Emperor may have represented. Napoleon may have found a ready ally in the country to which he just sold Louisiana. Britain would likely have targeted Louisiana for conquest as it was "illegally" taken from Spain (which was France's ally at the time).
It is interesting to think that this potential "colonial" conflict with Britain may help stem some larger campaigns later. With a little distraction by an American alliance, Napoleon may not follow through on his worst mistakes of his reign like the invasion of Portugal and Spain. Austria was already tired of getting it's ass kicked, Russia , Denmark (certainly), Sweden, etc were more enemies of Britain than France near these times, especially after the imposition of the continental system, which may or may not come about.
1x1, yes, Britain beats America fairly handily if they don't have any distractions. With France as America's ally...
Note that Britain probably accepts peace at the status quo when/if war breaks out in Europe again. 1803-04, I think, were the only real times of peace in Europe for 25 years.