Nice job so far!
Screwless cover = rare earth magnets.
With a forstner bit, drill a very shallow recess in 3 or 4 places underneath the cover, and glue in some rare earth magnets with superglue. On the body side, drill holes a little bit larger underneath where the magnets will touch the control cavity ledge, then screw in a small screw into the bottom of each hole (use screws that attract to magnets), I'm talking a screw size similar to a pickguard screw. The magnets will be attracted to the screws and hold the cover in place - if the holes for the screws are a little larger than the magnet diameters, this will take up any misalignment between magnets and holes, and allow you to remove and replace the cover more easily than perfectly-tight fitting holes.
Adjust the depths of the screws so that the magnets *just* touch the heads of the screws, and the cover sits nice and flat.
HOWEVER - if your wiring is pushing up hard on the underside of the cover, this may not be enough to keep it in place, in which case you need to just go with a screw-down cover. Looks like you should be OK in that department though.