OPSEC time. On the plus side, Asuma is pretty much in the loop wrt our story. We can afford to play along with Hidan and Jashinism and he won't get upset at us because he knows that we were forced into it at scythe-point. All Asuma wants out of us is that we direct Hidan in a useful direction.
The ANBU listening in on the conversation? Less so. Asuma might have sent them away to create better OPSEC conditions for Hazou, but I wouldn't bet on it. Being in the same room as two Akatsuki is dangerous enough without cutting yourself off from the outside world. If this meeting turns violent, Asuma needs ANBU around to support him and go fetch backup. Which means if we lean into the Jashinism stuff, we're putting ourselves into a corner that Asuma has to back us out of.
Overall, not the worst. The same reason Asuma accepts our Jashinism would make sense to the ANBU when it's explained to them, so as long as we keep in that narrow zone of "playing along with Hidan while still being a true Leaf-nin at heart" we should be in the clear treason-wise.
From the other side of things, we need to continue leaning into the Jashinism to keep Hidan happy, but he probably won't be testing us much because he knows that we also need to look loyal for our Hokage who is in the very same room as us. Kakuzu, of course, doesn't care one way or the other.
So that's our general demeanor, but what about the meat of the conversation? We need to convince Hidan to track Akane down, without letting him know that it's a matter of X-risks. In this fiction, Asuma is far less invested in the outcome than Hazou is. Asuma cares deeply that such a promising Leaf-nin got attacked, but wouldn't be so concerned as to hire the Akatsuki in a desperate effort to find her.
(Note: Akatsuki isn't above this kind of mission. They've done ordinary missions as mercenaries for years before the Battle of the Gods, and tracking people down sounds like the sort of thing they might have done before. What's unusual about this, from their perspective, is that it's coming from Leaf (who Akatsuki has generally refused to work with in the past) and specifically Hazou (who has a personal connection to Hidan).)
With Asuma not able to visibly care as much as he actually does, the burden falls upon Hazou to sell the narrative. Hazou's pulling strings, calling in favours, and Asuma's going along with it because he does sympathize with Hazou's grief. That's a story we can sell, that lets Asuma press his case without revealing too much. Or rather, it's the part of Hazou's perspective that would be most productive to lean into, because it is not only true but valuable for us to have Hidan believe it.
Asuma has his own angle to press, though, and that's the potential AMITY violation intertwined with the incident. He'll want to take point in the conversation as much as he can afford to, for a variety of reasons, so we should expect him to focus on that once Hazou's established the main story and motive.
From here, as things enter into the negotiation stage, it becomes harder to predict. Will Hidan be up for the mission? Will he want something from Hazou in return, like having us dive deeper into Jashinism? But at that point the OPSEC concerns are mostly resolved one way or the other, so we can simply proceed with the goal of maximizing the odds that we find Akane and/or whoever attacked her.
