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You can trap them by filling a shallow container with 1/2" of cooking oil then set the rim of the container at ground level. The earwigs will be attracted by the cooking oil and drown.
You can also sprinkle diatomaceous earth around the areas where you see them. You can get this product at any garden center. It's non-toxic so it won't harm your dog. Diatomaceuos earth is fossilized diatoms which are tiny needle like animals. They are too small to harm people or pets but just big enough to spear the earwigs.
The earwings seem to be getting under the cover for the BBQ, and in around the enclosed tank area.
The 2 options I mentioned are about the only things you can do short of insecticide. The diatomaceous earth won't hurt your grill. Just keep it away from the burners. It's a powder like substance. You could also try setting the traps near (even on) the grill