God is said to be πνεῦμα (pneuma) in John 4:24 but is he ever said to be a "soul" ψυχή (psuché) in the Bible?
God is spirit, and those who worship him must worship in spirit and truth.
John 4:24 (ESV)
The Greek word ψυχή (psuché) is the root of the English words "psyche," "psychology."
Is God ever said to be a "soul" in the Scriptures?
There's a lot of parts to your question. It confuses theology proper with anthropology without explicitly showing the connection. Additionally, you mention a Greek word, which seems to preclude the Old Testament. Regardless, you might find this verse interesting which includes the word "נֶפֶשׁ/nephesh":
Isa 42:1 -- Behold my servant, whom I uphold, my chosen, in whom my soul delights; I have put my Spirit upon him; he will bring forth justice to the nations.
There are many other verses where the Lord speaks of having a soul. It really comes down to whether or not you are willing to take them anthropomorphically. Even your typical trichotomist would deny that God is composed of parts though there may some other sense in which our being is analogous with his.
