That's an ambitious plan for a week.
I can list a bunch of places to see and things to do here in the mountains, but you'll scarcely have time for a short jaunt (budget 30 MPH) on the Blue Ridge Parkway which runs perpendicular to your path to the beach.
I'm not saying the beach ain't pretty, and Charleston isn't a Charming Town but maybe your lady would enjoy a night or two in a Floating House on Lake Fontana instead:
You could also visit some waterfalls in the area:
Asheville has a delightfully high per-ca pita number of breweries to explore with flowing taps and affordable accommodations within comfortable walking distance:
If you need bike servicing, gear or advice on routes, you can always stop by Acme Cycles, a valuable regional riders resource.
If you're a little saddle-sore and your hands are too numb from riding to massage your lady, you can always enjoy one of the very delightful twisty roads to Hot Springs, NC for a soothing soak:
Western North Carolina (WNC) is a renowned summertime destination for outdoor enthusiasts of all stripes: Backpacking, rock climbing and mountain biking opportunities are afforded by a plethora of outfitters scattered throughout the region. Rafting, kayaking and canoeing outfitters also abound if you need a little cooling off.
Please note that at this current time there a number of road closures caused mostly by rock slides and the (don't laugh) brutal winter we just survived. These closures include Interstate 40 at the NC/TN border, US 129 (Tail of the Dragon) at the NC/TN border (the amusement section of it is open daily from the NC side). Sections of the Blue Ridge Parkway remain closed hopefully only until mid-May. One section between Hwys 276 and 191 may be closed well into summer due to unstable ground.
Stay in touch, if I have some time I wouldn't mind showing you some of the more scenic and challenging back roads in my neighborhood (Henderson/Transylvania Counties).