Is it permissible to use the phrase "the justice of Heaven," and does "Heaven" in the verse (And in the heaven is your provision and what you are promised) mean that it provides for us?

If the speaker intends "justice of the heaven" to mean the justice of Allah Almighty's Law from heaven, then there is no objection to that. Indeed, it was narrated in the hadith of Umm Ayman that she said: "I weep because the revelation has been cut off from heaven." As for Allah Almighty's saying: "And in the heaven is your provision and what you are promised," the provision here is rain, by the consensus of the exegetes, and what is meant by heaven is clouds, for everything that is above you is heaven, and rain is the cause of provisions.