I'm not sure though if we've ascertained whether you are having a front or rear routed control cavity? If rear (as Woks is assuming), then you'll be pushed to get a 3-way switch in there, especially with an inset rear control cover plate plus enough depth in the top to safely mount the 3-way switch without the wood being too thin. Allowing 36mm depth, plus a 1.5mm flush cavity cover, only gives a 1.5mm wood thickness for mounting the switch to. Very precise routing required. You probably couldn't go thicker than 3mm anyway if you want the switch's knob sitting on the thick part of the lever, rather than looking a bit odd sitting higher up the thinner part.
A front-mounted control cavity would work better with this body thickness.