What about the MRP in postwar France? I guess in a way it arose mostly because the national-conservative option was discredited by the Vichy experience.TL;DR
When Catholicism has a limited socio-political relevance, it has incentives to operate through traditional political channels of the right (and left) as in France.
When it is socio-politically dominant, it will probably ally or be a core element of the conservative-national-Catholic alliance on the right, like in Spain.
Sweet spot: Socially influential but politically weak encourages it to organise itself.