What is your objective?
Do you have time to change your mind and looking for different directions to go?
An UL barrel will shoot as good as a heavy if its made with the same care and quality. A thin barrel will heat up faster and may cause a POI shift sooner.... My old factory thin barrel would shoot only 2 tightly then the 3rd would begin to move. A slightly heavier barrel might shoot 3 or more as the weight goes up before POI shifts from the heat. A carbon barrel is just a pencil thin barrel wrapped up, but don't scratch, ding or chip it... I don't see much practical weight advantage over a quality thin steel barrel. I'm uncertain how fast they heat up and cool down.
Flutes are only good for reducing the weight of a heavier barrel (and not by much...), might as well go thinner if you want an UL rifle. Flutes otherwise do not provide any heating/cooling advantage as its all relative to total surface area. It would be nice to know how much adding flutes reduces weight but I cant imagine enough to make a practical difference over the next size down. They do look attractive and can add that final touch to a build that makes it stand out.