The easiest way would be to leave the drywall in place and add rigid foam insulation on top. Use at least 2 layers of Styrofoam insulation, stagger the seams, and be sure to seal the corners tight. Sheet rock and Batt insulation will mold if condensation forms on either of them. Use caulk and spray foam on every gap, crack, or seam to prevent air from leaking through.
The R value of the existing insulation counts when you're building a cooler in a corner of an already insulated room. Add enough insulation to equal R25.