Sherem’s response to Jacob’s rebuttal turned out to be ill-fated. He could have retracted his allegations, but a retreat would have been hard, because the laws were pretty strict against false accusers. People could not lightly initiate a complaint without its being, in effect, a sworn statement. Presumably, if Sherem had retreated, Jacob probably would have not pursued the matter, but as you read in Deuteronomy 19, the punishment for those who initiate false lawsuits was strict. "Then shall ye do unto him as he thought to have done unto his brother." If someone falsely brought a claim of capital significance and lost, the punishment was that the accuser was put to death. The threshold for bringing a legal action in this world was very high. They did not have a litigious society as a result of that.
So, at this point there were Sherem’s accusations and Jacob’s testimony in contrast—and it was a tie, a stand-off. The most common method of breaking such a tie was to draw on divine directions. In the case of Jacob and Sherem, it was Sherem who said, "Show me a sign." He asked for some kind of divine oracle or manifestation, and that put the burden back on Jacob. Jacob had brought into the discussion the Holy Spirit, and therefore Sherem no doubt felt justified in saying, "Alright, let us take this one step further. You show me something to prove that God is on your side." It appears that Sherem genuinely believed that he was right. He was about to learn that he had been deceived. I am willing to give him the benefit of the doubt; this matter was so serious that he would not have just gone casually through this as a political or intellectual maneuver of some kind.