It sounds like you might be talking about the Big Lynn Lodge?

I did a bunch of google searches on motels in (or near): Mt. Mitchell, Little Switzerland, and Burnsville, but didn't find a whole lot; certainly not anything I would call "close by". The only things I could find were some "dude ranch", and the Celo Inn.

I called the Celo Inn, and wound up talking to some gent (the proprietor I suppose - I didn't ask or get his name) who turned out to be a fellow cyclist, one who has himself biked the entire BRP before. He told me he was completely booked up, but said that if I got down in the area, after biking all day from the Millers Creek area, and couldn't find anything - to give him a call, and he'd come pick me up. I would hope to not have to need to take him up on the offer, but I thought it nice gesture - one fellow cyclist reaching out to another.

Anyway, he said there was another place in the area, one that was even closer to Mt. Mitchell than his plae - Albert's Motel, so I booked me a room there for the night. The rate was reasonable, and it's not far off the Parkway, and not too far from Mt. Mitchell.