Perfect Bayesian Equilibrium in a two stage game with incomplete information
In your example, you would still use backward induction to solve for the Perfect Bayesian Equilibrium (assuming the distribution of private costs has full support). In fact, the second stage of your example is similar to a Bertrand competition with asymmetric information. You can refer to the following paper for a general solution of the second stage game: 


Spulber (1995) "Bertrand Competition When Rivals' Costs Are Unknown", Journal of Industrial Economics 43:1-11. 


The first stage decisions should follow in a straightforward manner.