Here are my thoughts (but full disclosure, we have only sailed catamarans). I think you will be fine with all of the experience that you have. We have never used more than half of our gas (but again, we have 2 tanks) even while running the generator generously. So you may not need to refill which would remove the need to stop at a dock. Consider spending the first day with the check-out captain, perhaps end at Norman and arrange to pay for him to hitch a ride back to Nanny Cay. One day should give you plenty of time to practice sailing and work on mooring balls. You could practice docking on the fuel docks before leaving. Also, consider paying for Horizon's Express checkout, where they bring you in and refill the boat - this will give you plenty of time to come in on the 26th (from Cooper or Jost), around 9 and still make the ferry.

General itenerary could be:
Day 1: Norman (snorkel Caves and Indians)
2: Sail to Cooper - use boaty ball or get there early for FCFS balls.
3: Sail to Baths (or straight to North Sound - Leverick, Saba, Bitter end)
4: Another night in the North Sound - consider renting a car at Leverick - go to Baths from there, Savannah Bay, Coco Maya (lunch), Hog Heaven, Nail Bay, etc.
5: Sail to Anegada (I would not miss it). Be diligent with instructions on sailing there. Consider renting a car. Defintely hit Cow Wreck/Tipsy's. Then Lobster dinner that night at one of the many great places.
6: Sail to Marina Cay - Go to lunch a Scrub - let kids enjoy pool, beach toys etc. (consider getting a slip here for the night, particularly with the kids), or just dinghy over from balls at Marina Cay.
7. Motor to Monkey point for a snorkel. Consider a quiet night at White Bay Guana Island. Or Go to Cane Garden Bay
8. Go to Diamond Cay/Little Jost (Sandy Spit, Bubbly Pool, Foxy's Taboo, B-line
9. Great Harbor. Spend day at White Bay (Soggy Dollar), dinner in Great Harbor
10. Early sail back to base.

If needed, refuel at Leverick, Cane Garden, Great Harbor. We always enjoy paddle boards. Definitely provision with Riteway or Bobby's online, and Tico or Caribbean Cellars for alcohol. They will deliver to the boat. You can get a few things at small riteway market at NC. Also restock at Leverick, Scrub (or Trellis), Cane Garden, Great Harbor.

I would consider using boatyball for Cooper and Great Harbor. Renport is expensive, but great highspeed internet - kids will like it for streaming, but you may not want that.

Your family will have the best time and again, you should be fine with your experience. Just get comfortable with boat operations and maneuvering with mooring balls with captain.