Unfortunately the author of the book sparingly used middle names, or middle initials and also referred to both Charles Sr.and Charles Jr. as “Charlie” in her writings.
I tried to find out if Charles Sr. even had a middle name or initial. Even old court documents did not provide a middle name or middle initial for Charles Beuchat Sr.
Dodie’s post above indicates that Charles Beuchat Jr.’s middle initial was “E.” but I could not find anywhere if the “E.” stood for Eugene or not. Maybe it does, as PAMM’s post indicates a couple references to a Eugene Beuchat. I really can't say.
With that said, I can comfortably say that the book does not provide a reference to a Eugene Beuchat.
The book, not having an Index, is a little more difficult to navigate, but going back over various areas I did find a reference to Lee (Beuchat), a son of Charles Beuchat Sr., who was studying medicine in St. Louis when Charles Beuchat Sr. was hospitalized there in 1923 to undergo an operation.
Also, with reference to my post above,
(Note: Livia, as I read, was Charles BeuchatJr.’s second wife Livia Toller. His first wife was named May.
and PAMM’s post,Beuchat, Charles E. ~~ Haskinson, Mary U. 18 Oct 1914 Las Animas County Colorado License #1175
I did find that book has a few spelling errors and the spelling of “Mary” may be one of those occurrences.
Sorry that the book couldn't specifically help in the research your family, and Eugene Beuchat.