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Byakuya is aware of Gin's bankai.

Byakuya at his best simply has better showings.

Gin sliced Hiyori in half which is whatever. He fought with Hitsugaya in a relatively non-serious bout but since neither used Bankai it is hard to say how he'd have handled it. He fought Shinji, which isn't that noteworthy since it mostly just seemed to be sword fighting and Shinji is surprisingly meh in feats himself. He then had a not too impressive showing against Bankai Ichigo, who was clearly still distracted by the Aizen stuff. Then Gin dies. I suppose it is logical to power scale him above Grimmjow since Tousen seemed to be and they were relative equals.

Byakuya curbs Renji pretty hard, destroying his Bankai in the process. He proceeds to have a much closer fight with Bankai Ichigo and might have only lost due to the sudden appearance of Hollow Ichigo. Most importantly for this fight, Byakuya while heavily injured reacts to Gin's Shikai to save Rukia. In Hueco Mundo he has a respectable kill on one Espada and helped kill Yammy. Two years later he would defeat Tsukishima, the most dangerous of the Fullbringers, and while As Nodt did defeat him he would later return to pretty easily curb 3 Sternritter offscreen. Byakuya was later one of the only 3 capable of stalling Gerard and if we powerscale him off of Rukia/Renji's improvement from training with 0 Division then there is a pretty strong case for him being second only to Shunsui(though knowing Kubo he's probably equal to Kenpachi).

Based on how they fight, Byakuya has a clear advantage since he is the only captain who really bothers mixing up his attacks with Kidou and his focus on Shunpo will allow him to deal with Gin's attacks. Most damning of all for Gin though is that his true strength, the ability to prey on the psychology of his opponent, won't really work on Kuchiki. Tsukishima's powers weren't enough and Byakuya did eventually come back to face As Nodt down, even if he did let the Rukia do it. Both of whom are way better at messing with their opponents than Gin. Gin will win with one attack because of the poison but Byakuya will not let that happen and his Bankai is much harder to dodge than Gin's.