Awesome idea!
As for the eventual demise of the colony/republic, i think that the janissaries in the ottoman empire are a helpful example. after all, they were professional 'guardians', raised from childhood, who were not allowed earthly goods, the brightest of whom would be sent to the 'Enderun' institute to learn how to govern (the similarities blew my mind when i first noticed). So our republics guardians could gradually become a powerhouse which stands against innovation or reformation to ensure its own existence. among the highest order you would have byzantine levels of intrigue & backstabbing for the top job. the rabble could rise up with the core revolutionaries being the underground poetry&drama scene (banned by the law) and guardian rejects.
as per the setting &POD, perhaps a different pelepponesian war (which socrates took part in OTL), leading to a different socrates who ends up defending the imagined republic (which he calls 'kallipolis'-'the beautiful city': cool ironic name for a dystopian timeline imho), rather than ending the thought experiment on noting why it's not possible. Plato would then be more like a John Locke, or Rousseau, and would be formulating what the city would actually be like, institutions, laws and all.
mind you that when you have such a text, the colonizers don't have to contemporaries or greeks; karl marx was a german but russians liked his ideas. so i don't know, a rich and eccentric roman can give it a shot. even the discoverers of the new world (who would have rediscovered plato around that time) can want to do it. think utopian socialists in early american colonization.
not sure about the butterflies of the slightly different pelepponesian war and the book though..