These are the only things I'm interested in. My personal opinions aside, I'm genuinely curious whether the government has a legitimate legal leg to stand on here. The reason I'm still replying is that none of those you've suggested have been terribly convincing.
Whether anonymous political speech by someone claiming to be an employee of the federal government but not acting in an official capacity is or is not a violation of the Hatch Act or any other applicable law (dubious) or any agreement they may have signed as a condition of employment is one question. Whether the fact that this individual has claimed anonymously to be a federal employee can compel Twitter to release their personal details on the basis of an assumption that either or both of the aforementioned factors (laws, agreements) will apply to them is another question entirely. While they're anonymous, how would you know which agreement they signed, if any? How would you know whether they are actually a federal employee?
"Tenacity" aside, this seems like the most plausible theory yet, but still a stretch. But I am not a lawyer.
edit: this update is relevant: https://www.washingtonpost.com/news/the-switch/wp/2017/04/07...