IMO one of the best triggers for this would be a period of crisis. For example if they lose quite a bit of land in wars (say 20% of the total number of provinces in 30 years).A fair compromise may be to keep the current system but only make a few AIs willing to westernize if the proper conditions are met.
Such big changes are more likely to happen during a period of unrest.
Or, alternatively, under enlightened leadership - perhaps a 5+/5+/5+ leader?