I'm sorry. I can't see the beetle well enough to make an ID.
While it's certainly possible that the larvae of the beetles chewed the holes in the leaves, I really can't say for sure. It could just as well have been the larvae of a butterfly or moth. To be perfectly honest, I'm not positive the hole in the leaf was caused by any kind of critter. As a rule, if something is eating the leaves of a plant, you'll find much more damage than is evident in the picture. Are there holes in several of the leaves, that just don't show in the picture?
On the other hand, damage from caterpillars/larvae is usually on the edge of the leaf, rather than in the center. A slug or snail will eat through the center of a leaf, so perhaps that's the real culprit. I have a lot of slugs and snails in my yard. I often find them during the day, hiding under the rolled rim of pots such as the one in your picture. You can set out shallow containers of beer as bait for slugs and snails. Don't place the containers right close to the plants you want to protect, though. Either bury the container up to its rim in the soil, or else give the critters a little ramp to crawl up to get to the beer. They drink it, get drunk, fall in, and drown. I have had excellent success using beer for slugs, but snails ... well, not so much. For snails, I prefer to pick them off by hand, drop them on the ground, and step on them. There are many ways to deal with them, however. Type slugs and snails
in the search box on the left of the upper part of the page. Click Search, and you will find a wealth of ideas for dealing with the problem.
It does look as if your pepper plant might have some powdery mildew on the leaves. A good organic treatment is to spray it with a mixture of 1 part milk to 10 parts water. Be sure to spray the stems and the undersides of the leaves, as well as the tops.
If you like, you can spray your plant with a solution of soap and water to kill caterpillars and larvae, but the treatment has to be repeated every 7 to 10 days, about 4 times, to kill any new ones that hatch out. I don't know whether it will have any effect on the adult beetles. Be sure to use soap, and not detergent. Read the label before you use the product. Mix about 1 tsp of soap to 1 quart of water. Spray as described for the milk solution.
Sorry to have rambled on so long, but HTH!
"Isn't it enough to see that a garden is beautiful without having to believe that there are fairies at the bottom of it too?" - Douglas Adams