Finding an optimal solution to a linear program among solutions of another
Suppose $P_i$ is as follows:
$$\min c_i^Tx$$
subject to $$A_ix=b_i$$
$$x \ge 0$$
Let's write down the dual:
$$\max p^Tb_i$$
subject to $$p^TA_i=c_i^T$$
I would construct the following linear programming problem
$$\min c_2^Tx $$
subject to
$$c_1^Tx=p^Tb_1$$
$$A_1x=b_1$$
$$p^TA_1=c_1^T$$
$$x \ge 0$$
$$A_2x=b_2$$
Note that I have used duality theorem to ensure that the solution for the new system is an optimal solution to the first system.