What is the ruling on buying by invoking blessings upon the Prophet (peace be upon him), such that the buyer says to the seller, "By invoking blessings upon the Prophet," and the seller gives him his request without financial payment?

If the needy person says this to the seller to make him understand that he does not have the money, so the seller gives it to him for free, then this is permissible and there is no harm in it. It is among the noble morals of the seller, although it is better to refrain from using this phrasing in this context.