You don't need to do anything. The switch will pick up ground via the screw heads of the mounting screws. (this "via the screw heads" trick does NOT work for receptacles).
If you really want to attach the ground wire, then find the screw hole in the back of the box that is tapped 10-32. That is specifically for a ground screw. You can't use screws tasked with some other purpose. You can drill and tap your own #10-32 hole, or you can use a grounding clip on the edge of the box. Of course you don't have to do any of these things for a switch or self-grounding receptacle.
Now, that thing you're doing with the wood blocks is not allowed. You cannot make a "box extension" out of wood, because the job of a junction box is to contain fires from electrical arcing. Wood would only accelerate such a fire.
I gather you have been using wood screws to mount the switches on the wood; instead use 6-32 screws to mount it to the metal. Further, you will need a metal box cover so the box is fully encapsulated in metal.
If you really, really need the switches to sit prouder than that, you can either rework the box with a taller "mud ring", add a Legrand Wiremold Surface Conduit Starter Box as a 1" box extension, or add something like a Hubbell PTC200GY switch extender, which is made to be a weatherproof outdoor cover for switches (but if you only use it as a spacer, I won't tell :)