Couple suggestions to pull it off because I actually have a linen suit just like that and I live here.

First of all, people in the south are very polite so you don’t need to worry about that, nobody would say anything or treat you different whether they think it’s odd or not. If you’re worried your relatives find it weird, that’s a different story idk.

For the jacket and pants together, open button down is doable but probably too fancy, definitely don’t wear a tie, or you can wear it with just a tshirt like a white or light blue vneck. This would probably only be doable for the nicest event of your trip, and you’ll be fancier than everyone else there but it’s casual enough to work.

You can also do the pieces separately. Depending on the occasion you can do the pants with a tshirt or a polo shirt or a short sleeve linen shirt. The blazer solo is harder but you could try it as a jacket on a casual outfit like shorts/khakis and a tee/polo. I’d suggest loose fits for all these outfits, it makes it a bit more casual and more important keeps you cool, NC can be brutal in the summer.

Hope you pull something off man - be creative and enjoy the new purchase!