The BH trip runs most days, as you suggest. The operator which goes most often is by far AquaDives, followed by Amigos del Mar, Ambergris Divers, and Ramon's. AquaDives have by far the biggest boat, and for a trip across the open ocean biggest is usually best. The day comprises a dive in the BH itself, followed by a wall dive off Half Moon Caye, lunch and time for a brief walk around on HMC, then a final wall dive just off Long Caye. The HMC wall dive is commonly regarded as one of the best dives in Belize.
The only other day trip regularly run from AC is to the Elbow at the southern tip of Turneffe Atoll. Because there is much less demand for this trip it maybe only goes once a week or so, so you need to start looking for a trip as soon as you arrive. Like the BH trip, it's only actually confirmed the previous afternoon, when enough divers have signed up and paid. Typically the minimum is 10.
Day trips to Glovers, Belize's third atoll, don't run from AC as it's too far.
Although you didn't ask this, the BH trip is also suitable for snorkellers, but the Elbow trip isn't.
Obviously local dives are available all the time, weather permitting. Typically boats leave at 9am for the first dive, return to dock for the surface interval, and go out again at maybe 11am for the second dive. Afternoon dives generall follow the same pattern, with the first dive starting at 2pm. These times can often be varied, forwards or backwards, so long as the boat will be available and all the divers agree. We often start morning dives at 8am, and indeed even earlier is good as well - it's bright and sunny by 6am.
The atoll trips generally leave at 6am, and you are generally picked up from your hotel dock at 5:30 (in a smaller boat). If you're further afield pickup may be earlier, whilst if you're really close to the dock of the operator you can walk there, to be there by say 5:45. All this is dealt with during the briefing when you book, the previous day.
Where would you be thinking of staying?