Yeah. Been said before but bears repeating. A kithrak would NEVER let someone else claim such a prize. It's the weapon everyone is looking for. His pride would demand that he take it himself and present it to Vlaakith.
I think if SH is not in the party, the kithrak should read your mind and thus know you don't have it on you. So he and the dragon fly off to go to your camp to get SH. After you fight the patrol, scene shifts to camp. You play SH and those at camp as you try to escape the dragon and Voss, dodging into a ruins or portal or something.
If SH is in the party, you learn she hid it at camp. The rider and dragon take off to get it. You fight the patrol but learn that the weapon jumped from SH to you before leaving camp. You just didn't know it.