‘A “king” who goes against Torah can be overthrown.’
Not al pi halachah. Rechovam was a rasha, but it is clear from the text of Tanakh and the rabbinic writings that Yeravam ben Navat should not have rebelled. The Naviim are blistering in their criticism of the kings of Israel and Judah, but how many times did they call for a popular rebellion to overthrow those kings?