Even if the toggle bolts work for reinstall, you still need access to remove what's there, if a bolt spins you're back where you started.
Going through the aft quarter bulkhead has the advantage of not risking a deck leak, but it has the disadvantage of turning this into a two-person job. A temporary cutout near the work means you can do it yourself.
If you build a pocket shelf just make the floor angled down, with drainholes in the facing, as they've done. Not sure it's worth the time and effort depending on the cost of one like that pictured. Marine consignment stores often have that sort of thing fairly cheap.
Those pocket shelves are a great place to toss a winch handle ... (remember? you're in the winch world now
) or the odd bit of line, a handheld GPS, VHF, etc etc.