How would you have government without ideology? FOCJ and panarchism would be the closest to such a thing, allowing for everybody to voluntarily participate in their own ideal form of government, but they are both ideologies in and of themselves and the latter isn't necessary a form of government but rather a lack thereof.
"How can you favor the Separation of Church and state without also accepting the Separation of Ideology and State, unless your motives are duplicitous and hypocritical?" Because people accept god out of blind faith, its disengenious to use this force them into supporting men who are obviously unrelated to god.
People accept ideaology out of reason, and mabey accept or reject them as they see fit, without risking burning in hell.
There is no moral imperative in the lives of the faithful to follow secular ideology, or philosophers.
Thats the point. To prevent the abuse of people, by levering their faith for political gain. Coversly, it protects churches, because it removes the imperative for the political corrupt to join to manipulate people politically.