Why did he try a second time? His employer, King Balak, had sent another delegation to him to convince him to utter the words of curse.
Did Bilam expect that a high-ranking delegation would change God’s mind? Bilam was no fool and he knew in advance that God would still refuse him permission.
Was he trying to appease Balak and tell him that he had done his best?
Maybe his problem was with himself, not with the king. He really wanted to curse Israel, not that he had anything against Israel as such but because he was a mercenary who did what the highest bidder paid him to do. Yet he was not without a sense of conscience.
Even as a mercenary he was worried about outraging his ethics. A tortured personality who could not tell himself outright that something was wrong and should not be done even if it meant he lost some money as a result.
Jewish tradition calls him “Bilam the Wicked”. His wickedness was in his lack of moral courage.