If the counter overhung the cabinet opening a little as it is supposed to you could use sink clips like these. Given your unusual opening with no overhang you would have to make your own clips. Since it's a drop-in sink I would attempt to just use silicone and see how that goes.
If the silicone isn't enough, you could try cutting little slits in the sides of the cabinet to accept the clips, using a rotary tool.