Here are some data for you, expressions which are similar or synonymous whatnot:
Judith 9.12: 12 Hear, O hear me, God of my father, God of the inheritance of Israel, Lord of the heavens and the earth [δέσποτα τῶν οὐρανῶν καὶ τῆς γῆς], creator of the waters, king of all thy creation, hear my prayer!
Wisdom of Solomon 18.14-16: 14 For while gentle silence enveloped all things, and night in its swift course was now half gone, 15 Your all powerful word leaped from the heavens, from the kingly thrones [ἀπ᾽ οὐρανῶν ἐκ θρόνων βασιλείων], into the midst of the land that was doomed, a stern warrior carrying the sharp sword of Your authentic command, 16 and stood and filled all things with death, and touched heaven while standing on the earth.
Psalms of Solomon 2.30: 30 He is the king over the heavens [βασιλεὺς ἐπὶ τῶν οὐρανῶν], judging kings and rulers.
2 Maccabees 15.23: 23 So now, O sovereign of the heavens [δυνάστα τῶν οὐρανῶν], send a good angel to carry terror and trembling before us.
3 Maccabees 2.2: 2 "Lord, Lord, king of the heavens [βασιλεῦ τῶν οὐρανῶν], and sovereign of all creation, holy among the holy ones, the only ruler, almighty, give attention to us who are suffering grievously from an impious and profane man, puffed up in his audacity and power."
4Q530 (4QEnGiantsb ar), column 2, lines 16b-19a apud Florentino García Martínez & Eibert Tigchelaar: 16b The ruler of the heavens [שלטן שמיא] came down to earth, 17 and thrones were erected and the Great Holy One sa[t down. A hundred hun]dreds were serving him, a thousand thousands 18a [were ...] him, [and ten thousand times ten thousand be]fore him were standing. And behold, [book]s were opened and the sentence was proclaimed.
The Dead Sea scrolls also have:
Lord of the heavens
host of the heavens
sons of the heavens
watchers of the heavens
seat of God in the heavens
YHWH in the heavens
heavens of his kingdom
war of the heavens
word of the heavens
Mishnah, Berakhot 2.5: 5 A bridegroom [who married a virgin] is exempt from the recital of the Shema from the first night until the end of the Sabbath, if he has not consummated the marriage. It happened that when Rabban Gamliel married he recited the Shema on the first night. So his disciples said to him, “Our master, have you not taught us that a bridegroom is exempt from the recital of the Shema?” He replied to them, “I will not listen to you to remove from myself the Kingdom of the Heavens [מַלְכוּת שָׁמַיִם] even for a moment.”
Wisdom of Solomon 18.14-16: 14 For while gentle silence enveloped all things, and night in its swift course was now half gone, 15 Your all powerful word leaped from the heavens, from the kingly thrones [ἀπ᾽ οὐρανῶν ἐκ θρόνων βασιλείων], into the midst of the land that was doomed, a stern warrior carrying the sharp sword of Your authentic command, 16 and stood and filled all things with death, and touched heaven while standing on the earth.
Psalms of Solomon 2.30: 30 He is the king over the heavens [βασιλεὺς ἐπὶ τῶν οὐρανῶν], judging kings and rulers.
2 Maccabees 15.23: 23 So now, O sovereign of the heavens [δυνάστα τῶν οὐρανῶν], send a good angel to carry terror and trembling before us.
3 Maccabees 2.2: 2 "Lord, Lord, king of the heavens [βασιλεῦ τῶν οὐρανῶν], and sovereign of all creation, holy among the holy ones, the only ruler, almighty, give attention to us who are suffering grievously from an impious and profane man, puffed up in his audacity and power."
4Q530 (4QEnGiantsb ar), column 2, lines 16b-19a apud Florentino García Martínez & Eibert Tigchelaar: 16b The ruler of the heavens [שלטן שמיא] came down to earth, 17 and thrones were erected and the Great Holy One sa[t down. A hundred hun]dreds were serving him, a thousand thousands 18a [were ...] him, [and ten thousand times ten thousand be]fore him were standing. And behold, [book]s were opened and the sentence was proclaimed.
The Dead Sea scrolls also have:
Lord of the heavens
host of the heavens
sons of the heavens
watchers of the heavens
seat of God in the heavens
YHWH in the heavens
heavens of his kingdom
war of the heavens
word of the heavens
Mishnah, Berakhot 2.5: 5 A bridegroom [who married a virgin] is exempt from the recital of the Shema from the first night until the end of the Sabbath, if he has not consummated the marriage. It happened that when Rabban Gamliel married he recited the Shema on the first night. So his disciples said to him, “Our master, have you not taught us that a bridegroom is exempt from the recital of the Shema?” He replied to them, “I will not listen to you to remove from myself the Kingdom of the Heavens [מַלְכוּת שָׁמַיִם] even for a moment.”
I do not know of any instances specifically of "kingdom of the heavens" before the rabbinical literature (of which I have given only one instance of many) apart from the gospel texts in question, but if a "king of the heavens" holds a realm, surely that realm would be a "kingdom of the heavens," and I seriously doubt that the rabbinical writers behind the Mishnah got their usage from Matthew or other Christian texts.
As for the Greek, for my money, Matthew is probably just taking the Semitic term שמיא seriously. If he supposes that Jesus would have been speaking in Hebrew or in Aramaic, then he would naturally suppose that Jesus said "heavens" (plural) rather than "heaven" (singular).
You, Martijn, did something very similar when you learned that Matthew's original has the plural rather than the singular: you ignored the singular in many of the prevailing English translations and now rather write the plural when referring to Matthew's text. Likewise, Matthew may well simply be ignoring the singular in the prevailing Greek translations and rather writing the plural when referring to Jesus' words.
YMMV.
Ben.