I am absolutely against talking to Asuma about Ami.
He's our boss. He needs to be informed or we risk committing treason by omission again.
Our goal is to grow closer to Ami, not turn her against Mist as if she's just any Mist Jonin.
We have multiple goals. Our main goal is not dying. We do that by not getting thrown into a kill box. Ami
is a Mist jonin: that's what matters to the politics here and we can't ignore that just because it might make her feel sad.
Getting advice and orders from the Hokage and other Leaf nin would make this a political move, and I predict Ami would feel betrayed when she found out.
Ami just had a rant about Ren not making a logical play to include a savvy ally in organizing Mist. As a jonin she also knows that circumstances like this aren't personal: this is just how ninja business works.
I argue that we should treat this as a family matter instead, based on what Hazou and Ami both have in common -- Keiko. Ami is already considering accepting us as family, we just need to give her time to come to terms with it.
It's inseparably a family matter
and a political one. Unless Asuma says that we should not contact her again (which I highly doubt he would do as I said previously for Doylist reasons) we will most likely see Ami again with Asuma's approval.
Even Ami went through the legal channels to see Keiko in Leaf once Ami knew where Keiko was. Feelings should not sway us to commit treason again otherwise we will prove once and for all that we are untrustworthy and therefore a liability to Leaf. I'd rather not see all of the good we've done and could do go up in flames for the sake of keeping one foreign agent maybe happy. It's just not worth it.
Besides, it's not like there's any specialists on Ami psychology in Leaf.
What? No one specializes in psychology for one particular person. Leaf does have experience with abnormal personalities and the Yamanaka's data on primitive psychology which is probably more than anyone else we could find in a reasonable time frame.
And what do we expect Asuma to say to us?
"I'm glad you brought this to my attention since anything pertaining to Ami automatically makes the situation internationally relevant. Continue, but be careful and if circumstances change let me know. It would be a boon to Leaf if we could sway Ami to our side but she's such a loose cannon in the first place she would probably make you guys look as reliable as the sun rising each day. She was trained by Mari so she knows all of the tricks to make a mark but if there is anything else you might need or if you feel in over your head I'll have some people you can talk to for advice, unwinding, and debriefing in whatever combination works for you. Don't run yourself ragged over this issue by trying to tackle this by yourself and I'll expect those check-ups to verify if she is getting to you. You are not in this alone and hopefully together we can convince Ami she isn't either, not if she does not want to be."
He'll tell us to continue doing exactly what we've doing anyways, which is bond with her and sympathize with her I'm hopes of turning her against Mist, when Ami has literally no actual loyalty to Mist in the first place.
The act of talking to your leader about what you are doing is necessary in any military organization because there is always a chance that the unexpected actions of a subordinate could ruin the larger plans of said leader. This is such a crucial organizational concept that there is a word describing the actions you seem to want to take: insubordinate. Insubordination, not following the chain of command, is very
very bad. Unchecked insubordination leads to these types of conversations:
"That's not your decision to make," Asuma said with forced calm. "It's the Hokage's. Or do you think every clan head should have the right to go over the Hokage's head when they think they know better?"
"But Hyūga—"
"The Hokage might not be the wisest or most experienced leader in the village," Asuma said. "Recent times are proof of that. But you know what I have, and Hiashi had, that you don't? A village of experts ready to assemble on command. There would have been a Clan Council meeting. There would have been a discussion, determining the village's official policy on Akatsuki, and our finest diplomats and negotiators would have written Leaf's letter. That is how you're supposed to do this. You don't just write whatever sounds right. First contact is precious, and you've poisoned that well for us."
"It wouldn't have mattered," Hazō said. "I took action exactly because there wouldn't have been any of those things—because Hyūga would have dismissed the idea."
"As would have been his right. The Hokage, and only the Hokage, gets to make choices that could put the village in danger.
"I'm not claiming Hiashi was a good Hokage for that sliver of time he was in charge. I'm not claiming I liked him or his policies. But do you know the word for when you ignore a Hokage's authority because you don't like him or his policies?"
"Treason"
Then we get thrown into kill boxes. Since we have already had this conversation with Asuma he will probably send us to trial before pulling the trigger.
Ami only cares about Keiko, herself, and maybe Hazou in that order.
Great. This is one factor in the political ramifications but it
does not justify committing treason to protect her feelings.