Does the ballcap mod convey the batter up perk after the book was read?

[SenpaiThatIngnoresYou]

So the mod that turns helmets into ballcaps, does that allow the batter up baseball hat perk to kick in? Also does that perk need a ballcap AND a coat to get the 5% damage reduction or is it 5% reduction per clothing item worn?

 

And is it the same case with the magnum enforcer suit buff? Does that need both or is it 5% per item of clothing?