I live in Nashville (and my sister lives in Germany). You should be fine. It's a great place to visit.
Any place in the world has the small chance of a drunken idiot making a fool of themselves by revealing themselves to be racist. We're no different. But you're not going to be harassed by police officers or anything like that (unless you're doing something worthy of their attention anyway).
The only places where I'd be the slightest bit concerned would be in the small town rural areas between Nashville and the national park. Even then you should be fine, but the odds of running into a racist idiot will go up from like 1% to 3%. And even if you do, it's not like you'll be in danger. They'll just do their best to make you uncomfortable by saying something nasty. They won't refuse you service or anything crazy like that.
Honestly, even in those areas you're likely to be looked at suspiciously because of your foreign accent more than the color of your skin.
But 1) I sincerely don't think he sees it that way and 2) it doesn't weaken Putin's position. I don't honestly believe Trump is a direct asset, but I 100% believe Putin has people closer enough to pull strings and plant ideas with a man who is very clearly open to any suggestions if they are pitched with a populist and/or 1%er bent.