I think there's still room for miscommunication-rather-than-malice-or-gross-incompetence. It's a crisis situation, the info and documents are being passed around internally, probably in a panic, someone forgets to include agreements about not sharing outside of legal in an email to another member of legal, who is under pressure to figure out something to do, and fast, and so on.
The only way their response would be termination in the described situation is if they genuinely had no interest in protecting the privacy of the talents. Even if it was a crisis situation, this would not be the response.
If they genuinely believed that Doki's document was a public statement that was going to be released if she was terminated, and that it was actually doxxing the talents, they would not just terminate her. That would be calling her bluff and daring her to release the public statement and publicize the private information of their talents.
There is no way they would do that in an industry where even talking about previous public personas is looked down upon, unless they didn't give a single fuck about any of their talents. It would genuinely be worse than what actually happened, even if they were just malicious little shits to Dokibird (Which they are). Because then they wouldn't be callous malicious shits to just people leaving the company; they would be callous malicious shits to everyone who is in the company, and it would be even more of a reason for everyone to graduate ASAP.
I know a guy who knows a guy who knows a guy who knows an asshole who has been in a situation where one party has threatened to go nuclear during a negotiation and tank the entire thing multiple times. That asshole has been on the party that threatened to go nuclear and the party that received the threat. That fucking asshole stated that every time the party he was on threatened to go nuclear, he was worried that the other side would just say fuck it and the deal would blow up, but usually they would come back with some kind of concession to salvage the deal. Likewise, whenever the other party threatened to go nuclear and blow up the deal, it would be an all hands-on deck situation to try and figure out some kind of concession to make to keep the deal from blowing up. The only thing that could make a deal blow up is if they were finally at the point where it wasn't worth it to make any concessions, and it was more worth it to continue fighting than to continue negotiating.
I cannot think of any situation where anyone with two working braincells in Kurosanji would think that it is more worth it for Doki to release info about their talents publicly than to continue negotiating unless they gave zero fucks about any of their talents, even their favorites.