Just to be clear on what the 2e manual says in regards to Unacceptable Orders and making someone die by them.
Exalted 2e, Core Rulebook Page 180
The default system assumption is that every character finds ideas that he cannot ("Suicidal despair," "Challenge the Kukla to a duel," "You will fly if you jump off this bridge") unacceptable. It also assumes assumes that these are the only unacceptable orders.* Persuasion can overcome any conscious belief but not the survival instinct. This is sometimes realistic but is explicitly not based on realism. The rule exists to empower persuasion while making sure that characters influence their enemies to do more interesting things than just "die".
*It notes after this exceptions, like Solar Motivations.
So, generally speaking it's true you can't use social-fu to get someone to die, at least not without a more concerted effort to destroy their sense of self-worth and will to live. But as with all thing in Exalted the GM is the ultimate arbiter of what the rules are, and the rules can be bent, changed, or outright removed, and it's fine so long as the GM is consistent and has good reasons it doesn't affect the fun.
Here I think that getting Kusakabe to kill himself through clever use of social-fu and UMI would be acceptable and interesting. No order to kill himself is given in the plan, but special circumstances make him choosing it a possibility that doesn't normally exist. There are actually up to three social attacks here:
1. Lelouch's speech + UMI -> Kusakabe is convinced he's dishonored the JLF and the Japanese people, playing right into the hands of Britannia. The notion of offing himself is strongly suggested, but still left up to him.
2. Kusakabe hallucinates the ghosts of his ancestors staring at him with shame, disappointment, and anger, should act as a second and totally unexpected social attack -> Kusakabe should be even more convinced of his shame and dishonor. (Or he might freak out. Should be interesting either way.)
3. If Kusakabe hasn't followed through at this point, Lelouch does another social attack to once more call Kusakabe a cowardly wretch + UMI -> Kusakabe feels even more ashamed. Then he sees Zero and gets inflicted with deep self-loathing.
Dishonor and self-loathing in and of itself wouldn't be enough to drive someone to suicide - this would never work on a Britannian, or even people from most other cultures. However, Kusakabe is a member of the JLF, which is inundated with Japanese warrior culture, and seppuku for deep dishonor is part of that. It only has a chance of working because of the culture. (which is also why in canon it was believed by Kusakabe's men that Zero was able to convince him and his guards to off themselves out of shame)
Interestingly enough, it should also be possible for someone to UMI Suzaku into doing something suicidally stupid - this is because it's actually his Motivation to die since he murdered his father. He would refuse any suicide like shooting himself or hanging himself, but anything that might get him killed in battle or saving other people he would readily accept.