i can't comment on STJ and i'm an anti-sleep aboard based on other peoples comments such as noise, proximity,but the postives for the kids may outweigh the negatives

i would think the peter island slip is ridiculously expensive, the one mooring ball is 65/night and it would take you about 4 minutes to be dockside, or you stay in great harbour, you can anchor for free or get a mooring ball and walk to peter island resort quickly and the kids will have lots to see, they can walk from great harbour past sprat bay, past deadmans over to big reef bay, a beautiful walk and lots to see and do, try picking up coconuts that have fallen on the grassy path near big reef bay and play coconut bowling, we had such a great time as a family doing this

another activity we enjoyed was hermit crab races, everyone collects a hermit crab, then someone draws a circle in the sand, about 1-2 feet in diameter and all the crabs go in the circle, first crab to crawl out of the circle wins, best spot to find crabs is in rocky outcroppings such as at sandy cay

we've stay on a saba rock mooring and never had problems as its another quick dinghy ride back and forth to BEYC or Saba

at Leverick, our preference is to contact Nick who's a regular on TTOL and have him arrange a car or a car and driver, the driver option might be good for the kids because generally the driver will add interesting local information and stories which may be entertaining, visit the baths from the land side, enjoy the freshwater pool at top of the baths after lunch, explore VG, ask the driver if the mini golf is still open, then return to leverick and let the kids enjoy another fresh water pool or the older kids could rent a kayak or stand up paddle board or try scuba diving, our kids and niece and nephew were the same ages when we took them diving with an instructor, or take a dinghy out to all the islands and do a beach crawl, moskito island owned by Sir Richard has a small beach jammed full of all sizes of conch shells, the kids would enjoy playing with these, immediately across from this beach, across the cut, is another tiny beach where the water is so shallow, the fish come up and swim around your feet and legs as long as you stand still, the longer we stood, the bigger the fish.

At Norman you have to moor but a great activity for the kids, aside from all the snorkeling spots, is to find the path ( or ask ) behind the restaurant/gift shop and climb to the top of the island, a very doable activity for these ages, and a fantastic view from the top, there's a surprise at the top for the kids which they'll think is neat, there's a helicopter pad up there!

Hope some of this is helpful, maybe not for dock vs mooring but hopefully for activities for the kids!