I was there last year on business, in the middle of the polar vortex so it was quite unseasonably cold. I'm going again for business in a couple of weeks. For these business trips, our VP insists we all bring our cowboy boots. I basically wore smart casual + cowboy boots. So jeans, dressy blouse, blazer and cowboy boots. It's been awhile now, so my memory is fuzzy, but I think I also brought a dress and wore it with cowboy boots. Basically what I normally wear, just with cowboy boots as a nod to Nashville. We did go out and about (dinner, pub crawl, bands, dancing etc) and I never felt out of place.
I'd most definitely bring your cowboy boots, or western-y cognac boots, because where else will you get to wear them? I feel totally out of place wearing them here in Boston. Which is a shame, because they are oh-so-comfy.
I think that Anthro dress would be super cute with cowboy boots. Or booties, as shown in the pic. Booties would work just fine too.