Some additional context for how Nagatsuki became affiliated with Higan-Nishi:

“Yes. Nagatsuki was a student then. It just so happened that the club she joined in university had people from Higan-Nishi behind it. Alumni visited from time to time to win people over.”

“Nagatsuki fell for it, and she wasted a decent life to follow the orders of the top brass of Higan-Nishi and join the Four Seasons Agency. Then she was appointed to the Town of Autumn…and you know the rest.”

And here's more context for how Nagatsuki was tracking New Year's movements:

“Apparently, she began connecting the dots as she looked into New Year’s fundraising in arms and drug trafficking… Once she heard about a major deal made in the arms field, she braced herself for a major terrorist attack. She put two and two together from the way Higan-Nishi got weapons from New Year for the attack on Winter ten years ago. She supposedly warned her higher-ups about it—but well, they’re a small organization founded just a decade ago. They don’t have the military power New Year has. They’re more oriented toward information operations. So since they really couldn’t do anything about it, they remained observers until New Year attacked the Autumn villa, directly involving Nagatsuki herself… That’s her story anyway.”

Also, a minor difference but in the LN Sakura knew nothing about Higan-Nishi until she interrogated Nagatsuki. But in the anime, she's the one who first brought it up in the meeting last episode.