I subsequently found another midrashic source discussing this same incident with slight variation.
Tana d’Vey Eliyahu Zuta 8:
Achav (Ahab), King of Israel, inquired of Eliyahu the Tishbite, saying: ‘It is written in your Torah: ‘Guard yourselves lest your heart seduce [you], and you go astray and serve other gods and bow to them. The Almighty’s anger will flare against you and He will stop up the heavens and there shall be no rain, and the earth will not give its crop…’ (Deuteronomy 11:16-17). Yet we serve idols and the rains do not stop! Rather, see how many good things have come to me!’ …Immediately, Eliyahu was filled with a great anger against Achav. He said to Achav: ‘Empty one! You have despised Him Who created the entire world for His honor and Him Who gave the Torah for His honor! By your life, I judge you by your own words!’ As it says: ‘Eliyahu the Tishbite, of the inhabitants of Gilad said to Achav, ‘As the Almighty, G-d of Israel, before Whom I have stood, lives, there shall not be in these years rain or dew except according to my word!” (I Kings 17:1).
Here there is no mention of the two meeting while visiting Chiel during his period of mourning. Furthermore, and more notably, while the passage in the Talmud (at least according to Rashi’s understanding) implies the continued rains hindered Achav’s idolatrous agenda, here Achav gloats of the benefits he’s enjoyed despite challenging the Torah’s promise of retribution for idolatry.
Rimzey Esh, commentary to this midrash, explains that even the seemingly wicked actions of these figures were perpetrated with lofty intent. Achav wished to achieve the highest form of teshuvah (repentance) which must be completely self-motivated, not fomented by outside stimuli. A lower level would be teshuvah motivated by a prophet’s rejoinder or fear of the Torah’s promise of punishment. A yet lower level would be teshuvah motivated by witnessing punishment befall another. The lowest would be teshuvah motivated by punishment that befalls the sinner himself. Achav thought the Almighty would withhold the punishment for idolatry from him so that he could repent in the most ideal way. For this reason Achav refused to repent following the death of Chiel’s sons so as not to diminish his repentance by having it follow another’s punishment, and Achav likewise scoffed at the Torah’s promise of punishment. Eliyahu responds by bringing direct punishment upon Achav by withholding the rain indefinitely, forcing Achav to diminish his repentance to the lowest level, in a way playing “chicken” with Achav, seeing how long it would take him to crack. Eliyahu calls Achav “empty,” for Achav believed his actions to be lofty while his thought process was indeed empty of sense.