I expect someone will come and add in some of thw war cries recorded by Josephus for the zealots, but there are some war cries recorded in the bible. Joahua 6:16 says joshua told the people Shout! For the Lord has given you the city!. Unfortunately it doesnt say what they shouted.

Later in with Gideon in judges 7:17-18 gideon does give a specific cry, we read "When I and all who are with me blow our trumpets, then from all around the camp blow yours and shout, ‘For the Lord and for Gideon.’” though other translation go "the sword of the LORD and of Gideon."

Going theough the rest of judges, the samuels and kings i find no more, but there is a lot of related informarion that you might expect of battles, namely the speeches given before hand or the curses given on the enemy, such as davids words to Goliath. Then there are victory chants as well. All of these seem more common than battle cries.

In retrospect its about what i see in war stories from other similar cultures, where the narrative is generally more significant than the battle cry, and the words of leaders are remembered.