‘ And Jephthah sent messengers again to the king of the children of Ammon.'
Jephthah did not expect for one moment that the king of Ammon would give way. Nor was he arguing a legal case. And he no doubt had the message read out to his own army before sending it. It was written as much for them as for the enemy. The aim of both armies was to put themselves in the right and justify their claims before their deities so as to be sure of their help, and to stir up their armies to support a ‘righteous (in their eyes) cause'. No one fights better than the man who fights for a patriotic principle and considers that he has a grievance and that his god is with him in it. And they would want their gods to fight for them.
So Jephthah was not only trying to put the king of Ammon in the wrong, he was also demonstrating to Yahweh why He should fight for Israel, and demonstrating that Melek and Chemosh had no good reason for fighting for Ammon, indeed that it was Chemosh who had given away the land in the first place. (Compare for a similar attempt at disillusionment, although on different grounds, 2 Kings 17:18-36).
The words he used show a good knowledge of history. This may partly have been a result of guidance and coaching from the elders and priests of Gilead, but he had grown up in an important family and would be aware of the history of the past which exalted Yahweh. But the essential message was his, for he knew exactly what he wanted to do. He wanted to put his enemies in the wrong, disarm their gods, and take away the sense of the patriotism of their action. Whether he really believed in their gods is beside the point. His recipients certainly did.