B&C is at the Beach Club, but the entrance is not inside the hotel itself. You have to exit the main hotel and walk along the sidewalk towards the Yacht Club. It is behind the pool, Storm a Long Bay.
The burgers are very good as is the ice cream. It's a small place so it tends to be crowded. There are tables, a few booths and a counter. Outside there are tables and chairs to eat at, too. There's a take out counter where you stand in line to get ice cream and sometimes food.
When the poster above said it was closed, I think they were confusing it with Ariel's, which is near by and has been closed for years. and no, you cannot make ADRs. There is an arcade next door to send time .. and $$ while you wait.