I might have misinterperated this but after the battle at Falme Byar "rode without looking back, he had seen all that he needed to see". I think that this indicates that Byar did not see the whole battle so Bornhald might have survived. Also it said that Byar had more important news about the figures in the sky, So this might mean that Byar was watching Rand fight and not the battle so again Bornhald might have escaped. Here is what I think. I think that Bornhald escaped and made his way back to where he was seen on the barrel. He was sitting disconsonately because he was thinking back on Falme and on the horrors that had befallen him. Who had more reason to be sad, seeing the legion destroyed.