Apologies for the delayed reply.
I have mine depicted very much like other city-state hoplites. Having said that, my Peloponnesian army is for the earlier 5th century BC around the years leading up to an including the Persian Wars - so there's not a lot of difference between my Perioikoi and Spartiates anyway. I don't have many cloaks but would incline away from red - leaving them for the Startans proper.
If your army is from the Peloponnesian War (lambda shields for the Spartans?) I'm thinking Perioikoi would not have the city state insignia but who knows really?