Bill - You'll find that 7 days is just not enough time, but you'll be able to cover a lot of distance with the PC, and especially if you are willing to get going early in the morning. In mid July, sunrise is 5:50 am and sunset is 7:00 pm.

Here's my suggestion:
Sun: Assuming you are able to depart by Noon, head to North Sound with snorkel stop enroute.
Noon - 1 pm: 8.1 nm to George Dog or Great Dog
1 - 2 pm: snorkel
2 - 3 pm: 5.9 nm to Vixen Pt, Prickly Pear Island
3 - 5 pm: dinghy to beach
5:30 pm: moor at BEYC or Saba Rock

Mon: 15 min AM motor to Leverick Bay. Suggest to dock so you can quickly top of water tanks before you depart.
Rental truck to Hog Heaven (if planning to eat, stop here on the return trip).
Continue to The Baths (bring NPS permit).
On return, quick stop at Coppermine Mine Pt.
Dinner at the Restaurant at Leverick Bay (if opened in July)

Tue: 14.2 nm to Anegada (2 hrs by the time you're moored or anchored). Get underway early to maximize your beach time.
Rent truck from D&W or Anegada Reef Hotel.
Lots of opinions on favorite beach bar, but I would stop for a drink at Cow Wreck, then have lunch at Anegada Beach Club.
If you want to snorkel and the swell is down, continue mid-afternoon on the north shore dirt road (if permitted by rental company, otherwise back track along south road and past the airport) to Flash of Beauty.
Return south through Settlement and back to anchorage.

Wed: 25.9 nm to Diamond Cay, JVD (<3 hrs). Again an early start is recommended.
Dinghy to Foxy's Taboo, then short hike to Bubbly Pool.
Mid-afternoon visit to Sandy Spit or Sandy Cay (30 minutes to hike around the Cay).
Continue 3 nm to mooring or anchor in Great Harbour, JVD. Dinner ashore.
Top off water in GH, if needed.

Thu: AM dinghy to White Bay, JVD.
(You can then continue the 5.5 nm to Cane Garden Bay and spend the night there. I would skip CGB, and have more time to explore Norman, Peter, Salt Island area).
Motor 10.0 nm (1~1.5 hr) from GH, JVD to The Indians
(Stop enroute in Sopers Hole for provisioning at Harbour Market, if needed, or souvenir and gift shopping, if desired).
Snorkel the Indians, continue to The Caves for snorkeling. Moor at The Bight. Drinks at Willy-T.

Fri: 7.2 nm (<1 hr) to The Rhone, Salt Island. I wouldn't rush to get there earlier than 10 am for best light. If you arrive early, anchor off the settlement and dinghy ashore to explore the salt pond and graves.

In the afternoon, backtrack 2.5~3.5 nm to Deadman's Bay (anchor) or Great Harbour (mooring), Peter Island, or continue 1.5 nm to Manchioneel Bay, Cooper Island.
Ashore you have dinner options at Peter Island Resort (dinghy to Deadman's Bay beach), Oceans 7 Beach Club in Great Harbour, and Cooper Island Beach Club.

Sat: 30 minute (4~4.7 nm) to Hodge's Creek Marina