My understanding is that the Aero and BA Stainless match barrels are one in the same. Aero has owned BA for several years now.
I've never had a Aero marked barrel but have and had plenty of BA barrels and they were all great. As good as any other top tier "commercial" match barrel. The only way you will beat them is a custom hand lapped cut rifled barrel with a matched bolt and it isn't going to beat it by much.
I would put BA a notch below White Oak for sure. With white oak consistency will be much better across the barrels. I have been reading some not so great things about the BA barrels as far as quality control goes lately on arf.com. I would still buy a BA barrel for a build, but only if on a budget. I would pick white oak over BA every time