There are numerous good places in that neighborhood (including Kokkari). Let me just mention one of the many possibilities: One Market.
1. It's right across the street from the Hyatt.
2. They have a Michelin star but still have some reasonably-priced menu options.
3. It's a huge space, so they should be good at accommodating groups without too much of the noise that might be associated with a smaller space.
4. Their menu should have a lot of universal appeal and includes a number of items that are marked as vegetarian or vegan.
Now, it might not be my first choice for a meal, most of the time, given how many other good restaurants there are in that area, but I've always found it to be pretty reliable when it comes to food and service and the ambience is nice.