That helps. If in the fall, I'd recommend mid-Sept - mid-Oct for warmer weather. Mid-Oct or later for cooler weather and fall colors.

Plan a full day in Asheville just for Biltmore. Fall is a popular time for visiting Asheville, so weekdays would be preferable for smaller crowds. You might even prefer 3 nights in Asheville if you want to get out for driving along the Blue Ridge Parkway and taking some mountain hikes. There are several small towns worth visiting. Some nice waterfalls, too. (Though they're not huge like in some of the big parks out west.) I agree that the Four Points is a very good option here.

If in the spring, the most flowers will be in bloom usually starting around late March - early April in Savannah/Charleston. Columbia usually starts about a week after Charleston. Asheville's probably a week or two after Columbia. It can all vary from year to year, of course. For Asheville, here's Biltmore's blooming calendar.

In Columbia, Riverbanks Zoo and Garden is worth a visit. Congaree National Park has Friday night Owl Prowls -- reservations required. The new Sheraton Columbia Downtown is a boutique hotel in a renovated "skyscraper" built in 1913. (And it just dropped to Cat3.)

In Charleston, consider tours of Fort Sumter, Fort Moultrie, and/or the USS Yorktown. Savannah has prettier downtown parks. The water shuttle from the Westin Savannah across the river is very convenient. I won't talk more about these two cities here because they're pretty well covered by guidebooks and online guides. The two cities are pretty similar in a lot of ways. While in Savannah, consider a meal outside at the Crab Shack on Tybee Island. (Good -- not fancy! ) In Charleston, we like the improv shows by The Have Nots.

The Westin Hilton Head is right on the beach. Visit the Coastal Discovery Museum for some nature tours. You could just spend a day here and then continue on to Charleston. IIRC, the Westin Hilton Head lets you use their facilities (pools, beach access, etc.) when staying at the Westin Savannah.

The Starwood properties in Charleston are about 15 minutes from downtown, but the Sheraton is good and will save you Starpoints as a Cat2. If you'd rather be downtown, you'll have to look for a non-SPG place -- perhaps a B&B. (A B&B in any of these 3 cities (SAV, CHS, AVL) would be nice.)

Edited to add:
I completely agree with sophiegirl's concerns about trying to cram too much into one trip in such a way that you'd not have enough time in each city. 7 days is too short for all 3 cities, IMO. I guess that's more along the lines of my thoughts regarding the driving distance concerns. If you could line up a 1-way car rental with unlimited mileage and no dropoff fees, could you fly directly to SAV and out of AVL or CLT? Are you planning to pay cash for your flights or use FF miles?

Yes, Beaufort, too! If in early October, perhaps the Shrimp Festival!