I really don't get why the Federalist Party would have dissolved in this timeline, having won the elections of 1812 and 1820.
The Federalists have been treading a path similar to that of the OTL Whigs for some time now: playing the role of the anti-Republicans rather than building up their base, failing to foster a new generation of leaders, relying on a handful of popular figures (one might call Marshall the Henry Clay of alternate history) to keep the party alive. Yes, they won in 1812, but that was against a remarkably weak Republican candidate running in the shadow of an unpopular administration (and I might add that, four years later, they were swept out of office by one of the biggest landslides to date). The fact that their nominee in the last election wasn't even a member of the party says it all: the Federalists have nothing more to contribute to American government. With their catastrophic defeats in the 1818 and 1822 Midterms, the few elected Federalists remaining in office slipped into the Republican fold, where they formed alliances with the "National" Republicans led by Adams and Clay.
List of Federalist Presidential Tickets
1796: John Adams/ Thomas Pinckney (F) 22% popular votes [LOST]
1800: Alexander Hamilton/ Charles C. Pinckney (F) 32% [LOST]
1804: John Marshall/ John Jay (F) 25% [LOST]
1808: John Marshall/ Rufus King (F) 48% [LOST]
1812: John Marshall/ DeWitt Clinton (F/DR) 74% [WON]
1816: John Marshall/ John E. Howard (F) 39% [LOST]
1820: Unpledged Electors (F) 13% [LOST]*
*Voted for DeWitt Clinton (I), who narrowly won.