What is the ruling on the statement of one who declares a persistent sinner an apostate if they deem the sin permissible, and obliges them to return to Islam by uttering the Two Testimonies, performing ghusl, and repenting? And was this statement transmitted from any of the Salaf?

There is no necessary connection between persisting in sin and deeming it permissible. Persistence might be due to weakness of faith or the overwhelming power of desire. A Muslim does not exit the fold of Islam unless persistence is coupled with mocking Allah or believing the sin to be permissible. Otherwise, it is by this (mocking or believing it permissible) that one commits disbelief, not by mere persistence. Ahlus-Sunnah wal-Jama'ah (the People of the Sunnah and the Community) do not declare those who persist in sins to be disbelievers. Thus, a perpetrator of a major sin—who has not repented—will not remain eternally in Hellfire as long as they are a monotheist. However, they might enter Hellfire and be tormented therein for a period, then be brought out of it to Paradise. Persistence in minor sins turns them into major sins.