Far be it from me to try to interpret the master Mr Cohen's lyrics. He can make words mean what he wants them to mean. Think of his songs as poetry.
I will only point out that there are two ways to use a word in the philosophy of language. We can 'use' the word for its literal meaning, as we usually do. So "hello" means "Good Day", and we have sentences like "He said hello".
But we can also 'mention' a word - this is called the use-mention distinction. When we mention a word, we use it as a "signifier", or a symbol. It doesn't just have its literal meaning, but is also a metaphor for an entire class of things. So "hello" can also mean "a greeting", and we can have sentences like "They kissed their hellos".
In many of the lines of this song, Mr Cohen is not only using but also mentioning Hallelujah. So Hallelujah can mean "Praise Jah", but it can also be mentioned to mean "A word or song of worship". E.g., the broken king composing a song of worship.
That's how he employs the words grammatically. As far as what he means: who can say but him?