A different way to phrase your question is to ask for sufficient conditions under which the Nash equilibrium for a price competition game yields the same payoffs under simultaneous versus sequential choices.
A partial negative result is in: E. Gal-Or, "First Mover and Second Mover Advantages", International Economic Review, Vol. 26, No. 3 (Oct., 1985), pp. 649-653.
Suppose that the game is symmetric and has a unique symmetric equilibrium under simultaneous choices: then the players' payoffs are the same. This paper shows that, under sequential choices, this is never the case when the best replies are increasing (the Leader earns less than the Follower) or decreasing (the Follower earns less than the Leader). So in this simple context, you would need to have non-monotonic best replies to achieve the desired result.
On the side, let me suggest that you add more context to your question. It seems you are implicitly assuming unique equilibria (how are we supposed to compare multiple equilibria?) and it is unclear whether $\pi_1$ and $\pi_2$ should be supposed to be symmetric.