What was the answer?
Malbim: Indeed, every Yisrael should want to avenge the blasphemy without reward 1 . The reward is for one who shows such Gevurah to kill this Gibor, for his Gevurah alone 2 (even if not for the resulting benefit).
This is like Malbim explained David's question, why rewards could inspire someone to fight Galyus more than avenging the blasphemy. However, salvation can come from one who is not so spiritual, e.g. Gid'on, Yiftach and Shimshon, who are called Kalei Olam (Rosh Hashanah 25b). Perhaps Sha'ul offered the reward for such people! (PF)
Ta'anis 4a says that Sha'ul said improperly. Perhaps a slave or Mamzer would strike Galyus - would Sha'ul give his daughter to him?!