About "Cry me a river."

There is a song "Cry me a river"(1953).
How should I analyze this title grammatically?

My opinion:
Cry(vt) me(Indirect Object) a river(Direct Object).
This sentence has a parrarel construction to:
He shed me tears of joy.
=He shed tears of joy for me.

"A river" here is a metaphor for "tears like a river" or "a river of tears." And "cry" here is used as a transitive verb like:
They lived a happy life.
He died a miserable death.

But some say, "a river" here functions as an adverb like "Thanks a lot." and others say,"Cry me a river" is the same as "Cry [for] me a river. "For" was dropped due to some musical purpose/reason.

In dictionaries, I can find "cry" as a transitive verb.
However, I can only find an example sentence like:
He shed tears of joy.
I cannot find any example sentences with the SVOO construction.

Please let me hear your opinion regarding how I should grammatically analyze "Cry me a river."

Thank you in advance.