Hawkman insists in Black Adam that there are only heroes and villains; there is no middle ground. It is surprising that Hawkman and the rest of the Justice Society cooperate with Task Force X's commander given Waller's morally dubious behaviour that tends toward evil. It's probable that Waller understood no member of the Suicide Squad possessed the strength necessary to defeat Black Adam.
Perhaps Hawkman considers cooperating with Waller to be a lesser evil. Hawkman doesn't consider Black Adam to be a hero, so he must be a villain, and in the eyes of the Justice Society, that means he needs to be imprisoned before he causes harm to anybody else. It's possible that Hawkman thought collaborating with Amanda Waller was the best way to accomplish that.