In some ways, Perdita`s wearing a disguise would have been a problem were she being portrayed as a Puritan. This play, though, is not set in Renaissance England, but in classical Greece, and therefore she is a pagan, and can behave in manners that a Christian Puritan would not.
The particular connection with Puritanism lies in the theory of art. First, she argues that natural flowers are better than hybrids, with the sense that nature is God’s work and thus better than human artifice. Her objections to artifice and imitation recall the aesthetics of Platonism, i.e. she is portrayed as a pagan Platonist. The Puritans adapted Plato (contra the Aristotle of the scholastics) as an important pagan philosopher and source of many of their own ideas; this Platonic influence on Puritanism culminates slightly after Shakespeare’s death in a movement known as ‘Cambridge Platonism’ that adapted pagan Platonic philosophy to Christian theology.
The setting is not Greece but the Kingdom of Bohemia and there is a hint of puritanism about Perdita since she won't paint: "were I painted", IV.4.101 and also she is reluctant to dress like a man and "disliken the truth of your [her] own seeming (...° I see the play so lies that I must bear a part....", IV.4.650-51.
Interestingly, the play was one of the 14 plays performed before the court on the occasion of the marriage of princess Elizabeth, James's daughter, shortly after the funeral of the death of his elder son, Henry. Elizabeth's future husband was to be the King of Bohemia. Elizabeth was nicknamed the Queen of Winter. So, the setting, the reference to both Sicily and Bohemia and the mythological regferences to ancient Greece are of paramount importance.