I don't like the BRP climbs out of Asheville. There's a fair amount of traffic, and it's hard for cars to pass cyclists with all the curves. Weekdays probably aren't too bad, though.
I drive up to the higher elevations and ride from there. It's a little complicated to make a route that has water or food stops available, there's not too many places on the Parkway. I like out-and-back routes, it's never boring to retrace my outbound route, and I see different stuff on the return, climbing the outbound's downhills, and with the sun moved across the sky.
Just one day
Only one day to ride? For me,
this route is the best one I've found on the Parkway. It's more climbing than you specified, but worth it. The climbs aren't steep, but there's no flat ground at all, so pacing and reasonable gearing is critical. The 50-55 mile version is probably comparable to 70-80 miles on flat to rolling areas.
I posted it in this thread post:
Richland Balsam to the Pisgah Inn route. Check out the photo album in the link. Carryout lunch at the Pisgah Inn, sit on the back deck and check out the views.
Starting at Richland Balsam, it's 46 miles, 5200 feet. (I don't like to start with a downhill, I'd rather extend the route a little and warm up with a climb.)
If you've got the right gears to set a moderate pace on 5-7% grades, you'll be fine on these big climbs. And I like this route that has somewhat shorter climbs with descents in between.