We see that King Chizkiya (Hezekiah) had a wicked father and son (Menashe). In fact the commentators credit Menashe with causing the destruction of the Bais Hamikdash by the length of his reign and the setting of the path from which the nation of Yehudah was never able to recover. In spite of that, we do not see the neviim calling for the ouster of the king and his replacement. The only revolt in the kingdome of Yehudah (after the split when Rechavam son of Shlomo took over) was against Athalia in order to restore the rightful king to the throne (after she tried to destroy the lineage of David after her husband died).
Note that I do not consider death of Yoash at the hands of his servants or the death of Amaziah as a revolt because in both cases it was done in order to put the proper heir on the throne. In the case of Amaziah, he was killed because the people realized that Hashem had forsaken them and allowed them to be defeated because of his sins. The navi did not come and tell them to revolt.
These kings had the full status of melech Yisrael as we see from the Navi. In any case, we see from other sources that the kings of other countries have to have their laws followed because of dina demalchusa dina. The kings of the Northern kingdom were at least on that level. They did not have the full status of "Melech Yisrael" because they were not of Malchus Bais David.
There is a medrash that Yeravam was offered the chance to stay a "royal house" if he would do teshuvah, but he turned it down.