The fundamental problem the Federalists faced was that they'd tied themselves firmly to an aristocratic social model that was already on its way out in the 1780s. The could have resonated in the South, which was still socially dominated by a landed gentry and would be for some time, except that it had become clear that Federalist economic policies favored Northern commercial and manufacturing interests.
In order to survive as a party, the Federalists would have needed to pivot to something very like the model that would later emerge as the Whig model: dropping the aristocratic social policies and extending Federalist economic policies to also appeal to yeoman farmers in the west and the emerging urban middle class. 1814 is much too late for this. I have a hard time seeing it happen later than the Adams administration.