I am getting the impression that a lot of posters are not appreciating just how serious and dangerous this meeting is. Any single word or action here could very well be recorded by history as a "the Archduke's driver took a wrong turn" moment.
There are mainly two reasons why I would prefer for Hazou not to mention it - aside from Ami likely already being aware of it:
1) IC he is still very much attached to his mother and might be conflicted about offering "her spot" to a stranger
2) What impression does this make to Ami if we are willing to throw our mother under the bus to get Ami to come to Leaf? This appears a lot more mercenary than we want to convey ourselves, especially since we know that Ami considers family important.
Does she? We know that she adores Keiko because of the type of person she specifically is and what she has done for her, but she was perfectly happy to manipulate and screw over her entire clan to prevent them from reassimilating Keiko's loyalties. The only other named character we know her to have a link to is known as the utterly ruthless "angel without mercy".
Distinctly not ignoring his feelings. Venting to someone about the huge amount of drama caused by his mom is totally understandable for a 14 year old. The entire point of this is Hazō isn't making any kind of offer. Ami will do all the leg work
While venting about the subject to a walking extrapolation engine who is already blackmailing his father and head of state is in character for Hazou, that's something which absolutely everybody has been trying to fix.
J is playing an extraordinarily delicate game with a frienemy moderate reformist Mizukage, the belligerent hardliner faction of Mist, all of the various self-interested clans in both Mist and Leaf, the leaders of the three other most powerful states on the planet, and a wildcard unknown faction who seems to have surprised everybody by just waltzing in and inviting themselves to sit down at the table to start blackmailing the most powerful people on the planet with Kurosawa skill and Kage-level talent.
Hana attacked and crippled the Hokage's fiance on her first night on the job, then physically assaulted and battered her, after which J, in cold blood, after ensuring that even his own ANBU wouldn't know about it, violated her diplomatic status by beating her to a pulp and threatening to disfigure her for life in a ploy to extract concessions from the Mizukage who very much will not want this information to get out either.
Again, J didn't even trust his personal ANBU bodyguards with knowing about this. Just a suspicion that
something happened would be enough to reorient the entire board in unpredictable ways, and that's without anybody taking that suspicion and extrapolating more from it. Moris are superhumanly good at extrapolating.
This is not a little thing. Even breathing the slightest shadow of a hint of the possibility of this would be summary-execution-level high treason. J would be less likely to kill Hazou if he just started telling Ami everything he knows about shadow clones. There are at least three different major world powers which would have an incentive to send an S-Class to murder Hazou just to keep him from mentioning it again. More would be incentivized to do it simply to reduce the number of potential surprises in the Great Game being conducted right now again.
Can't tell if this is supposed to be a joke or if you seriously imagined Hazou randomly vibrating on his chair while talking to Ami and then went "Yes, this is fine."?
No, I imagine him picking it up, saying "sorry, I have to take this", and politely excusing himself to talk to his hand about salt prices for a few minutes.
For me the point isn't to get Ami as ambassador. It's to get her to put pressure on Ren which will let Jiraiya get another angle to extract concessions from the Mizukage no matter who gets the position
Remember that Ren is the closest thing J has to an ally in these omnilateral negotiations. She has more in common with his goals than several actual
Leaf clans. Depending on what the actual objectives of the completely unknown third Mist faction are they could very well just use this information to get Ren 'retired' and replace her with a warmongering hardliner eugenicist. That's if it doesn't leak to any of the other major or minor nations who would be happy to do it just to sabotage the nascent Leaf-Mist alliance which is scaring everybody.
There is a reason that Hazou promised J that he would stay away from anything remotely sensitive or political. Hazou doesn't even have a seat at the table, and he'd be utterly over his head if he did. He is not in any position to try to be a player in this game. If something is not already ubiquitously common knowledge in both Mist and Leaf Hazou should not be talking about it to the Kage-level superhuman with totally unknown allegiances and objectives.
Anyways, gift: go or no go?
I think something minor would be best, given that Hazou and Ami haven't met before and this isn't an official meeting.
I'm thinking a small handful of explosives, because explosives are our thing so it's in-character for us and while it's not all that valuable or unique it has undertones that we care about Ami's safety and well-being.
Handing out explosive seals to foreign ninja without tower approval is also something we have been explicitly told is literally treason.
-and then there's the potential of the detail-oriented walking supercomputer noticing that Hazou's seals are all
exactly identical and inferring something both the Mist and Leaf royal clans consider a major secret.
-or, worse,
not inferring that and instead going down the "well, Leaf has printing presses and suddenly has so many seals that they can afford to throw them around like candy" route which leads to panic and a preemptive dogpile on Leaf before it can conquer everybody else with its world-changing technological breakthrough.
We noticed there's some valuable/rare lumber around in Mist at the moment, right?
We could give her some wood arts and crafts?
To elaborate: what if we have Hazou carve something small? It will be crude since he is no expert in wood carving but it could symbolize that a) he put some thought and work into it, b) that he is aware he is not a seasoned player at the political stage yet but that he is willing to give it a try and c) it is something he made completely by himself, not at all dependent on other factions, i.e. we come here to negotiate on our behalf and no one else's.
Look, if we're just going to ignore the unknowable potential misinterpretations of the subtext of a personalized or non-standard gift anyway, Hazou already has one (two) he isn't using anymore and we might as well maximize the number of people likely to kill him when they find out.
Play her
Tears of Red and give her the ruby pin.
Presumably we would be able to identify and find a good quality calligraphy brush.
I'm cool with doing some calligraphy, but handing her her actual name comes off a bit creepy in my book. Like, we haven't actually met this person yet.
Fine, if you're going to do calligraphy he also already knows something much more suitable than her name:
^_^