That changes things a lot, a family of 6 with children wouldn't be the best fit for liveaboard vacation.
Since you selected Cayman Brac over Grand Cayman then you are my kind of person! Although I actually like all 3 of the Cayman islands in different ways. I enjoy Grand Cayman but prefer not to stay at the pricey and touristy Seven Mile Beach and West Bay area, we like to stay in the more residential Bodden Town area. And I love both Cayman Brac and Little Cayman as dive vacation destinations.
I visited Bimini a couple of times on liveaboard trips and it used to be a sleepy, quiet village, but things have changed a lot since then.
It is wise to stay at a place where you can prepare most of your meals because eating out three times a day can get very expensive - and it wastes a lot of precious vacation time, IMO.
Personally, I am not comfortable with the idea of captive dolphin excursions, it seems cruel to me. I am also uncomfortable with the idea of teaching sharks to associate divers with food; although I did participate in a chummed shark dive off Nassau in my younger days, so I cannot condemn others for this practice.
If you really want Hammerheads and a Dolphin Swim then I think that Bimini may be your best bet, and you will just have to find the dive op and venue that best fits your needs and budget.
Just an FYI, I have done a couple of liveaboard trips in the Turks and Caicos and I saw lots of sharks in that area, but not hammerheads. I have never stayed on the island of Grand Turk but it sounds like a quiet, remote, funky place - but there is a cruise ship terminal at one end of the island and they offer family excursions including a "snorkel with the stingrays" opportunity.
Good luck!