Wasn't it because Jaime was being held by Robb? Tywin wanted one of his sons to do it for him?
Tywin assumed Jaime would not make it alive, which would make Tyrion his heir. He gave Tyrion some responsibility just to see how he could handle such an important task. And I doubt Tywin really wanted to drag Tyrion with him all the way around the Riverlands; he'd be more of an use at KL than with him.