A Slight Problem

I have a great deal of respect for Peter Navarre, and consider him the most interesting character in Lucas County's history. However, I've just found a story claiming he was descended from Henri IV of France. It seems a book, written in 1897, put forward a notion that Peter Navarre's family were somehow descended from Henri's illegitimate son Cesar, Duke of Vendome. Now, the Duke of Vendome had three children. His daughter Elisabeth married the Duke of Nemours. Her children are well documented, having been ancestors of the royal families of Italy and Portugal. His younger son Francois, Duke of Beaufort had no recorded children. Louis, the eldest son and second duke, had three sons. None of them produced children.
Just assuming there might have been an illegitimate child somewhere, who was only known to an obscure local historian of the late 19th century ( A tribe well known for being prone to flights of fancy. ), how would he have ended up with the name Navarre, instead of Bourbon, like the rest of the family, and why, having seriously wealthy, powerful, and childless relatives who had no qualms about recognizing illegitimate progeny, was he pissing around in 18th century Detroit instead of living the high life in France?