Is it necessary to wear the Silver Shroud outfit?

When doing the Silver Shroud quests, is it actually necessary to wear the Silver Shroud costume or can I wear my Power Armor or regular armor?

I know I can't get rid out of the outfit, but I don't see any point in wearing it.

No, but wearing it unlocks the "Speak as Shroud" option in dialogue:

Whereas any other armor just changes this to "Attack":

Other than that, the "Speak as Shroud" option is liked by some companions. I'm not sure which, but I've personally seen it happen with Paladin Danse (which also exclaims superhero-esque phrases when you attack any of the villains).