Hi Rob,
Sorry, didn't mean to cause offence, but I have about 6 or 7 CBA tutor books (French and Italian) and no two of them use the same treble fingering. Some of the older methods (pre 1970) were devised when the French musette repertoire tunes all followed fairly standard runs and chord formations, with a lot of them having been specifically written for 4 row CBA. In those days all of the French teachers forbade use of the thumb, although one or two permitted its occasional use, generally on the outside row only.
The Italians seem to have preferred the Anzaghi school where the thumb was used from the outset, and the modern French methods tend to follow suit, as the music has changed away from the more traditional stuff. I taught myself to play using the now old fashioned Ferrero method, and the other books I have were only bought out of curiosity. Out of the more modern methods I reckon Richard Galliano's is about the most practical. The thumb is used from the start and you'll be introduced to the 4th row early on. He does have an unusual way of playing 4 note chords, as he misses out the third finger in most of them. However, I haven't really looked deeply into the method. The good thing about his method is that he gets you working the basses early on and there is an accompanying CD, in case you are struggling.
I'm going to quote a professional CBA player who told me "If you get halfway through a Continental 5 row method, under no circumstances should you change over to another method, as it will set you back months if not years". He was spot on, as I went against his advice and spent years worrying about whether my fingering was right or wrong. These days I just play it whatever way I can manage it, as nobody's really going to care.
I appreciate you seem to be about to add CBA to your other instruments, so you'll already have acquired a good technique. I've never played PA or diatonic so I don't know what difficulties you'll experience.
At the risk of upsetting the apple cart again, there is no correct treble fingering for CBA. Each method will have its own version, although there will be some similarities among them. Perhaps the most controversial method book I ever saw was written by a French guy called Paul Chalier. Paul was a successful professional player and recording artiste, who reckoned that since most French musette players in the 60s only used three fingers, then he would devise a method which reflected that. His method book was short and sweet. Pinkies and thumbs were banned and his advice was to use as few fingers as possible when playing. Although this sounds strange, the early French musette was more of a folk type style and most of its exponents rarely if ever strayed into other music styles. Funny thing was that Paul Chalier was possibly better known under the alias of Pablo Caliero, as he was also a prolific player of tango music on the bandoneon. He actually devised his own playing method for that instrument as well, which was accepted without controversy (probably because it wasn't a CBA!). If you were to have asked Paul Chalier (before he died in 2002) what was the correct fingering for the scale of C, he would tell you that his was correct, but so would about a dozen others.
I assume there must also be more than one way to tackle a PA, but I've never heard anybody discussing it.
Good luck with buying the method book. I've tried to find it listed elsewhere other than at the museum, but have had no joy.