Imo, it wasn't unnecessary at all.
Suppose they start off with "He's not the real son of the marquis", what do you think the reply would be? 100% It would be along the lines of "if not, well, then who is he?"
To which they would have to tell the truth, because they don't want to commit double perjury. Given how cautious they were being, it seems to be something akin to fraud. The second reason for this set up is to force the lady to realize she's fallen into her own exact trap, and they can't sue them for the fact since it would be hypocritical.
Honestly he could have just sead that he was abandoned by his family (that would have been enough to disgust her) but him telling everything and provoking her and being like "do you still wanna marry him" was such a stupid idea cause if he thought for a minute he should have realized that she would take him just to spite him, and now not only did he endanger his best friend, he is also gonna make his sister miserable, for someone who was supposed to be smart he makes some idiotic decisions, i have to wonder how is he going to solve this ┑( ̄Д  ̄)┍
I can't get past chapter 13 without raging. This might be far fetched but I'm disappointed with the brother. I know the fake identity is a plan but "exposing" or telling Kiel's life in front of other people is not his story to tell. Instead of just saying that Kiel is not the real son of the marquis (?), I think the rest of what he said was unnecessary. Or maybe I'm just being too sensitive today....