These are merely my own thoughts.
The calves erected by Jeroboam in Bethel and Dan, were a Machievellian tactic to prevent the people from going into Judah for the Festivals, seeing the discrepancy between him and the Judaic King, and consequently rejoining the Southern Kingdom (Sanhedrin 101b). He also forbade outright the Biblical commandemnt to travel to the Temple for the Festivals. His son Nadab followed his father's example. Even his other son Abijah, who defied his father and allowed passage (moed katan 28b), was killed by Gd while he was still a lad, because Gd wanted him to die while he was still righteous, before he turned to the sins of His father (Zohar chadash 20a).
After Jeroboam's catastrophic fall from his righteous beginning, Baasa was Israel's last hope for relinquishing this political facade, and returning to the ways of Gd, as the third king of Israel. After him, Zimri seized the throne for seven days, and Omri and Tivni struggled for the rulership of a kingdom that had already seen a terrible trend of Idolatry from three different kings, a hazakah.
Perhaps Baasa's tribe was mentioned for this reason. I feel it is also significant that he came from Issachar, of all places, who tribe was defined by an expertise with and a love of the Torah and its laws (Yoma 26a, also the Midrash Rabah). Maybe the other ruling tribes were not mentioned because while they did nothing to rectify the machinations of their predecessor, Jeroboam, they arrived on the throne in a state of Idolatry, so to preserve their tribes' dignities for later generations, their origins were omitted.
Moderate support for this theory of mine can be found in Kings II 17, were Hosea, an Idolater himself, is deemed noteworthy for being the only Northern King to remove the guard posts and permit the people to go to the Temple (Gittin 88a). His tribe is not mentioned, because he worshipped Idols himself, but he is still alloted some merit for at last allowing Israel to follow the Torah and go to the Temple for the Festivals.