The three pop from urbanization cost no happy? Or you mean you can rush a circus and coliseum? Anyway, I found civil society and democracy the best policies for this when going for several large cities as opposed to one absurd city.
Best course of action with rush buy is to always rush buy food and growth buildings I think. For this kind of population challenge. What is required is fast medical labs. Maybe even aztec would ve strong for this challenge with their watermill food bonus.