Honestly probably a pound at max, there are no weight stats for the vessel that I can find anywhere. My guess would be probably under a pound. If you go by how carry capacity normally is the bird could carry 75lbs, but that seems...excessive so imma leave it up to you XD
Out of curiosity would the dex save be at disadvantage for Hawk if the bird was carrying Aura's Vessel? It's a small stoppered bottle