I always go to the Island dead on 7PM and stay until the entire basket is full of sharks, whale sharks, golden stags and horned hercules. Then I go back to the town, sell, then go back and do it all again. I normally stay until about 1AM when I get tired and go to bed You do have to have the night owl ordinance in effect to do this though. So yeah, like 7PM-morning sometime I'd say :3
The best time for me to catch beetles- mainly golden stags- on my island is from 8:30pm-9:30pm. As for sharks I find them during that time frame, but usually when I'm playing 11:00pm-12:00am I find two or three in the ocean at a time.