there existed state religions in the states in USA.
The USA in the games timeframe would be the only secular and tolerant religious state - as they themselves stated, e.g. in their treaty with the Pasha of Tripoli:
http://en.wikipedia.org/wiki/Treaty_of_Tripoli
At the federal level it was a pluralistic (better word then secular?) and tolerant religious nation.
But the federation consisted of states and the individual states could have a state religion.
Original 13 states with state church in the hole time fram of FtG:
Congregational - Connecticut
Original 13 states with no state church:
New Jersey
Pennsylvania
Delaware
Rhode Island
Original 13 states with a state church that ends in the time fram:
Maryland (1701/1776)
Virginia (1786)
Georgia (1789)
South Carolina(1790)
Massachusetts (1780/1833)
New York ?
North Carolina ?
None-secular without state church:
New Hampshire (Until 1877 the New Hampshire Constitution required members of the State legislature to be of the Protestant religion).
http://en.wikipedia.org/wiki/Separation_of_church_and_state_in_the_United_States
As there are no federation mechanism in FtG you need to choice between:
-State with vassals
-Some thing like Holy Roman Empire
-State without vassals
to represent USA.
And I think the current representation (state without vassals) is the best.
But if you should represent is historically right the individual states in USA should have different have different state religions: pluralism/secular, protestant (and reformed?) . In a similar way as the Holy Roman Empire gave there member states the choice to be Lutheran or Catholic after 1555.
But from a diplomatic prospective I assume that most of the leader of other parts of the world viewed USA as christian even if they at the federal level viewed them self as "secular". So to create the best historic diplomacy I think it should stay protestant.
How it was internally is impossible to re-create so I think what to create best historic diplomacy is what you decide in this case.