The Ingenium mentioned throughout the story is not Tensei Iida
Technically, the only thing to suggest that he is is the assumptions of the transplanted characters. However, such a positive alteration to the timeline seems out-of-place compared to the tone of the rest of the story, and Tensei is not the only hero to have used the Ingenium name. It's possible that he was injured just as badly or worse than in canon, and the Ingenium we hear about is Tenya carrying on his legacy. Hagakure is part of Gentle Criminal's crew
Hagakure hasn't shown up in the story at all, either as a cameo or as a character. She could easily have shared Tokoyami's motivations (possible spoilers?) for joining Gentle Criminal, but unlike Tokoyami, if she were to end up in one of Gentle's videos, she wouldn't be recognized easily due to her invisibility.