Prior to Aegon conquering the Seven Kingdoms, what do you think the Faith Militant was doing? They formed such a resilient and threatening opposition to the Targaryens, but we never hear them fighting other religious enemies before that. There’s nothing about them fighting the Ironborn in the Riverlands or against the North in their wars with the Vale. Did the unification of Westeros also improve the strength of the Faith, or is this simply a piece of history we’re missing?

Actually, we do hear a bit about them. The Faith Militant intervened on behalf of the pious King Humphrey of House Teague, when he attempted to suppress the Old Way and the Blackwoods revolted against him. 

But you’re right, it’s a missing piece of history. However, we need to be careful about where the Faith Militant might be active. For example, would they fight the Ironborn in the Riverlands? The Hoares were of Andal blood, had brought the Faith of the Seven to the Iron Islands, etc. If there’s not a religious cause, why should the Faith Militant support the Blackwood-supported Durrandons? 

Before unification, the Faith would have had to be very careful with secular politics – their interest is in spreading the gospel across Westeros, and the fact that they’re based out of the Reach is already going to make some kingdoms leery, so you need to show strict neutrality between the Kingdoms, unless there’s a religious rationale. 

Against the North, the Faith Militant may well have fought and we just haven’t heard about it yet. Now, they probably weren’t institutionalized yet when Theon “the Hungry Wolf” Stark stopped the Andals at the Weeping Water,  invaded Andalos, conquered the Sisters, and warred against the Hoares. But I would not be surprised if they were involved in the later stages of the thousand-year war.