What is the proper way to insulate a cantilevered bay?
In ascending order of cost, difficulty, and expense:


Bump-outs are notoriously leaky because of their complicated construction. Seal all air leaks in the applicable areas that you can find using caulk for small holes and gaps and expanding spray foam for larger ones. Cover any exposed spray foam with drywall or spackling compound to protect it from ignition, especially near a gas fireplace.
Remove any drywall or flooring materials in the area and insulate any stud bays that are currently empty or full of crappy fiberglass insulation with far superior Roxul mineral wool insulation, then replace the drywall and/or flooring.
Remove any siding materials on the outside to expose the housewrap or tar paper over the sheathing, and cover it with rigid foam or rigid mineral wool boards and re-install the siding. Note: to do this properly, please read http://www.buildingscience.com/documents/bareports/ba-1204-external-insulation-masonry-walls-wood-framed-walls/view?topic=resources/retrofits