"the Lance of Longinus" or at least the blade from the lance.
I think that secret is going to come out near the end of the book in spectacular fashion. Didn't Harry also bring out another item, the placard? I think those items, or even if Harry only uses one of those items; well, it will make quite a splash in both the Council and the rest of supernatural world. Even the Senior Council will first wonder how Harry got those items and then have to ask themselves if they want to cross Harry, knowing that he is even scarier than they realized. I can imagine at least one Senior Council member asking Harry where he got the spear and Harry answering, "I stole it!" Plus, Harry may have saved the day, so there could be that going for him.
I think the only secret which may cause Harry a big problem is his rescue of Thomas. To reveal that secret would put Harry on Mr. Etri's s**t list, a place Harry doesn't want to be. The alternative is to go along with the lie (illusion) that Harry and Lara were seen in flagrante, which will make the Council believe that Lara has her hooks into Harry.
Everything else; the Council doesn't really need to know.