How was God's reputation being threatened? Simply that while Israel deserved to be destroyed, if He were to destroy Israel (cut you off), then He would have broken his covenant with Abraham, Isaac and Jacob. Recall for example how in interceeding for Israel, Moses recalled to God the covenant:
Ex 32:11-13 But Moses sought the favor of the LORD his God. "O LORD," he said, "why should your anger burn against your people, whom you brought out of Egypt with great power and a mighty hand? Why should the Egyptians say, ‘It was with evil intent that he brought them out, to kill them in the mountains and to wipe them off the face of the earth’? Turn from your fierce anger; relent and do not bring disaster on your people. Remember your servants Abraham, Isaac and Israel, to whom you swore by your own self: ‘I will make your descendants as numerous as the stars in the sky and I will give your descendants all this land I promised them, and it will be their inheritance forever.’"It is interesting that God is concerned about how He is perceived even by unbelievers or foreigners. If this is the case then we should also be concerned about how other people perceive God through us. Are we upholding God's reputation? Do we give the impression that God is faithful?