Note that the stepper controller you specify can control "Up to 2 stepper motors (unipolar or bipolar) with single coil, double coil or interleaved stepping." Perhaps you will want a shield that can control four. Or you could get external stepper drivers (not a shield) and run wires to them.
If your LCD does not have an I2C or other serial interface you will likely run out of pins.
You have not said anything about what data appears on the LCD.