Can I chip in here? From AC there are several operators who go out to the atolls, including:-
Amigos del Mar
Hugh Parkey's Belize Dive Connection
Ramon's
AquaScuba
Island Divers
Only Amigos and possibly AquaScuba go with any regularity, the others go as and when. Hugh Parkey's have an excellent boat and I'd always check with them first. Island Diver's boat is a fairly standard 38ft (I think) part-covered skiff that is on the face of it not as good, but I have been out with them and we had a good ride and an excellent time. Once you know what you want to do I'd check with all of these.
Any of these operators will go to either Lighthouse or Turneffe depending on demand. As I said above, there currently seems little demand for Turneffe, but if you can get a group of 8-10 together they'll probably do it.
"The Aquarium" is one of the stops, the last one, made during the standard Blue Hole trip. It's very good, though my favourite is the one before, "Half Moon Caye Wall". These are all in Lighthouse Atoll. Turneffe Atoll is generally dived from AC as one of two alternative routes. Turneffe North is good wall diving, and is quite a short journey from AC. Turneffe South is much further, and it is at the far southern tip, as far as possible from AC, that you'll find the famous dive site "The Elbow". Hank here can eulogise on that site!
My personal favourite day trip is three dives at Lighthouse but excluding the Blue Hole. I used to run those trips, but it's the BH itself that is the big draw and it's difficult to fill a boat to go to Lighthouse and not visit the BH.