Come on, the basic pirate outfit is dark pants, a button up shirt, and sometimes a piratey hat. The rogue variant is pants, a matching cloak and a bikini top. Blackhat is some kind of goth porn fantasy outfit. The closest thing to colorful and flamboyant is the swashbuckler outfit, and that's pretty much shorts and a tank top. The boots and gloves are an odd choice, I'll admit. I can't see any of these raising an eyebrow with all the other crazy stuff running around out there.
The Zaxx mission is already restricting you to infiltration gear and no armor outfits, why do you want it restricted even more? And what wouldn't alarm the church? The bikini? Rags? Ganger? The pirate outfits are the closest things to normal clothes they have