Jake...
It does seem a bit odd - not exactly an American icon or patriotic image... Not even anything related to the sea!
My understanding from this and other forums is that the 1/96 is based on a Smithsonian model, which in turn is based on the famous "Hull" model in the Peabody-Essex museum in Salem. The Hull model shows the Griffin/Dragon on the billet.
Hull model photos:
i hull x.htm
The acknowledged expert in this area is CMDR Tyrone Martin. He states that John Rodgers commissioned the Dodge bros to render the billet work when he was in charge of her refit in New York and it included the "dragon" with the traditional scroll work.
tgmfigureheads.htm
Finally, I think you can see the Griffin on the Corne paintings commissioned by Isaac Hull to commemorate his battle with Guerriere.