I've looked at the BRNO site you provided a link to, and read the description of their process you provided. I'm not sure what they are actually doing, but it sounds like they are welding a damascus sleeve or chemise around a core of modern barrel steel: this would make a safe and useful barrel for modern cartridges, if properly done.
But, as a barrel maker whose associate in the enterprise is a metallurgical engineer, I have strong reservations about the suitability (read: strength) of any traditional damascus barrel with modern smokeless powders - it is a physical impossibility to make a composite of wrought iron and steel, with a structure based on a huge number of welds, (which may or may not be sound) which is as strong as homogenous steel.
The fact that damascus barreled shotguns can be smokeless proofed in Britain does not make me feel any better about the actual safety of such barrels: that any barrel passes such proof merely demonstrates that it could stand the stress ONCE. And, since few if any such barrels have been made in, probably, more than 75 years, virtually all existing damascus barrels are at least that old - they haven't improved with age. And even in shotguns, the average breech pressure with smokeless powders is at least twice that of blackpowder: every shot with smokeless loads is a proof load in a damascus barrel.
I've seen what happens to damascus barreled guns when they do fail: it isn't pretty, and in shotguns, they often fail at exactly the point where the shooter grasps the barrel(s), with the results you might expect.
And, really, damascus barrels are not even traditional in breechloading rifles at all. Just before his death, I was working with Lynton McKenzie (who was the Proofmaster at Birmingham) on re-boring some damascus muzzleloading barrels - damascus is not a good material for rifles if only because it is difficult to rifle with good dimensional control and surface finish (non-homogenous material), both of which factors are critical in making an accurate barrel. And who has any use for a rifle which is NOT accurate?
mhb - Mike