Well, overall quality I suppose. All the barrels are very similar profiles I'm just wondering if any of them would really make a difference compared to each other. I have a Sionics and it's been a great barrel. I've just heard a little about BCM's from Ted Procter about the lightweight barrels not wandering as much compared to other brands. I'm guessing he means heat stringing. I don't know if it's because of the profile or manufacturing process.
I think it's important to understand that you'll be dealing with 2 different sources of POI shift - heat and the suppressor.
The heat issue is directly related to barrel quality, or more specifically, evenness of the barrel walls down the length of the barrel. Most barrel bores are not perfectly straight and tend to wander, even if the ends are concentric, but a quality barrel maker puts a lot of effort into minimizing this. Quality of the steel is important too, and the better barrel makers will even reject certain batches of steel that aren't up to their standards.
This POI change will vary as you heat up the barrel, and may be exaggerated by the weight of the suppressor.
The POI change from suppressor mounting is partly due to gas effect on the bullet, but mostly because you're hanging a weight on the end of a steel bar (the barrel); that weight causes some deflection. That deflection is directly related to barrel stiffness, a long thin barrel will generally show more POI drop than a heavy barrel.
That POI change should be consistent and repeatable though, and is easy to account for with a mil/mil optic - just record your POI shift suppressed, and either dial that much when the can is installed, or hold on the reticle. The POI change (for example - 0.7 mil down, 0.2 mil right) is consistent at extended ranges; just add it to the values in your established dope chart.
Hope that helps.