When I think of the Federalists after 1800, I've always thought Hamilton. After Hamilton's death, I don't think of any important Federalists. If the situation leading to the duel doesn't happen or the duel is a gentlemen's shooting in air affair or whatever keeps Hamilton alive so he's the Federalists leader.
In 1804, with Hamilton dead, Jefferson faced token opposition. I think even with Hamilton alive, he doesn't face Jefferson as he realizes Jefferson will probably win.
In 1808, however the (Democrat-)Republicans are in trouble because Jefferson's 2nd term policies have been a disaster for the country as a whole. Thus Alexander Hamilton runs for president and beats James Madison for the Presidency.
Another founding father becomes President though in place of another founding father.