May-June begins the very best of the Florida diving season. Warm, clear and calm is typical. In addition to the bridge (you can time your trip and the dives it the way I suggested around any high tide at 8 to 830 in the evening), I would definitely consider one boat trip (2 dives) to the local reefs, which are very beautiful and usually have multiple turtle sightings during that time of year. Great to see them nesting at night, and then see them in the water during the dives!
+1 on staying on Singer Island. +1 for Brass Ring if you are a fan of great burgers. Another food idea, in my opinion the best seafood restaurant in the entire state (and I have eaten my way from Pensacola to Key West) is called the Food Shack in Jupiter, only 15-20 mins or so from BHB. Multiple species of absolutely fresh local fish, cooked in incredibly creative ways. It is gourmet but casual and the prices especially for lunch are super reasonable.