The fate of the woman has three potential outcomes:
She came to believe the truth of Solomon's ruling and faced the reality of her child being dead, and so she mourned the loss of her child.
Her baby was taken from her by King Solomon and given to another woman, so she came to terms with her child being given away, and so she mourned the loss of her child.
R. Hayyim Joseph David Azulai raises this question in his commentary to that verse and notes that he hasn't seen anyone discuss it:
לא ראיתי מי שדיבר למה נקרא מלך ישראל מאחר שהוא ממלכי יהודה ועיין במסרה כ"י דברי אמת כ"י להרב מהר"ש עדני ז"ל
However, there are in fact several earlier rabbinic authors that do address this. Radak in his ...