This relationship sounds almost perfect. Unfortunately, that "almost" is creating a problem you can't get past. Your man's secret time on chat sites is something you hate and refuse to live with.
I would like to know a bit more about why you have such a problem with chat sites, though. Are they sexual chat sites, or "Meet Singles" sites? Or are they just general interest or hobby-related sites? If they are sex/single sites, then you have every right to be concerned and troubled. If it's more for general chat and making friends, then I think you're being more upset than the crime warrants.
That said, regardless of what type of site it is, your man is doing something that bothers you, and he needs to stop. He knows you hate it, but he's still choosing to do it. That is a problem, and it's a big enough problem that you should consider breaking up with him.
Give him one last chance to explain himself. Ask him why he frequents these sites, why he doesn't care that you hate it, and if he plans on continuing. If you don't like his answers, move on. Almost perfect isn't enough for you, and that's ok. Wait for perfect if you need to.