We went to BB on July 4th year before last expecting the worst, but it was not too bad. We got there at 9:00 opening and rode the chair lift (my son loves that one and it can get backed up quickly) and slides that get lines first and saved the lazy river/bobbing pool for when it was more crowded. It definitely got busier earlier than other days.
I'm seeing that the calendar shows both parks opening at 10:00 that day. Not sure if this will change closer to the time, but if it stays at 10:00 then I would avoid it. That 9:00-10:00 was key for it working for us because it gave us an hour and a half or so before it got crazy.