When a charge (say positive) is placed between an upper positively charged plate and a negatively charged plate, it should experience a repulsive force from the top plate and an attractive force to the bottom plate, should it not? Then if I were to calculate the net electrostatic force on that charge, wouldn’t I have to double the magnitude of the force between the charge and one of the plates, since it is experiencing two forces? Why is this not the case?
Let me ask you: how are you characterizing the force between the two plates? Typically, the two plates are connected to a voltage source, and the force is found by multiplying the charge by the voltage, and dividing by the plate distance. The voltage already takes into account the charge on both plates, so there's no need to worry. If you're doing a more complicated problem where you look at the charge on each plate explicitly (maybe the charge on each plate is different, or something), then yes, you need to the charge on both plates into account.