Technically, the sefer doesn't stop using Achav's name, seeing as the division between Melachim 1 and 2 is a later Christian invention. However, the question of course stands for this chapter.
Da'at Mikra explain that the reason Achav is referred here as "the King of Israel" is because the main subject of the chapter is something of national importance to the Kingdom of Israel: The freeing of an Israeli city from enemy hands.
I'll add that similarly, in chapter 20, Achav is referred to several times as "the King of Israel", each time in relation to matters of kingdom-wide importance. Once when he submits to King Ben-Hadad (20:3-4), once when he seeks counsel from the elders (20:7), and three times during and after his meeting with the unnamed prophet who tells him he's going to fall as punishment (20:40-43).
Therefore, in a sense, we may say that chapter 22 is echoing chapter 20: Though Achav should be commended for also focusing on matters of national importance, ultimately, for his various sins, he was to be punished. He merited to bravely die in battle because he also cared about his people.