Sorry I'm late for this, but Miami-based here and figured that I'd chime in. I'l summarize/add to some of the good suggestions made above.
If you want to be *in* Miami/Miami Beach for good diving and little hassle, check out Divers Paradise on Key Biscayne. They are easy to get to, plenty of parking and the staff is amazing. They have two boats that run daily, so there are a variety of shallow trips available. They run out to Neptune pretty much every day. Easily accessible via a short Uber from Miami Beach or the Mainland.
As others have mentioned, Quiescence is a small private charter company in Key Largo. Small boat and A++ white glove service. Since it's your boat, you set the itinerary. The cost will be about the same as sending 6 people to any shared charter boat in Florida. This is about 1hr south of Miami and worth the trip if you have transportation.
As others have noted, there are no good shore dives in the Miami area. You would need to head north to Lauderdale by the Sea or the surrounding areas. Too much hassle, IMO, to drag the family 45 mins north, figure out equipment and tanks, etc.
Last comment: Yes, Miami is busy during that week. However, my experience is that dive charters are not crazy. I dive multiple times that week. Nonetheless, given that there are 5 of you, be sure to book well ahead of time. I'd recommend that on any weekend.
Enjoy.
