Using the Chomsky-Schutzenberger theorem to prove a language is not context-free?
here is a construction not exactly as requested but related/ somewhat similar. the contrapositive of the Chomsky-Schutzenberger theorem can be used to prove a language is not unambiguously CFL. it describes the close connection of the construction to generating functions that yields key insight into the problem.

see ORDINARY GENERATING FUNCTIONS OF CONTEXT-FREE
GRAMMARS / TANNER SWETT, EDWARD ABOUFADEL, sec 2.6 p 6