Three players of these curious instruments are profiled in the book: Pietro Frosini, Leon Sash, and Alice Hall. Frosini was the earliest of these:
Pic from this page: Pietro Frosini - Accordionist (1885-1951), which offers this detail about his instrument, including the proper term for it:
Always a player of the Chromatic Button key accordion, with a special bass system devised by himself, from the 1930's onward he used a Finto-Piano Accordion, which was really just a three row button key instrument, made to look like a piano accordion.
"Finto" means "fake," "false," etc. The Golden Age book has an excerpt of a more detailed profile of him, with a chart of his free bass setup, which was wholly unique. The Finnish Wikipedia page of Frosini has this chart of his layout:
A later player of fintos was jazz virtuoso Leon Sash:
Another was Alice Hall:
Hall used the finto at the insistence of her accordionist father.
There are examples on YouTube of all three's playing:
Pietro Frosini Rhapsody No3 in A Minor 1941
Leon Sash / Jazz Accordion / Pennies from Heaven
Alice Hall, "Pennies from Heaven," 78 rpm record (1949?)
Alice only made one 78 so it's serendipitous that she recorded the same tune as Sash, and we can compare their playing. Sash sounds like an even better accordionist but Hall's sides are really wild fun, with her scatting in tandem with her box playing.
Looking for pics of Frosini I came across clips of a modern player who is a disciple of him, right down to choice of instrument: Alf Hågedal - Robins At Sunrise (Pietro Frosini), Skei 2011. He plays a...3 row finto, I guess we'll call it. Both white and black keys on the piano section can be seen moving in tandem with buttons, so they clearly are functional in nature and not just installed to look "modern," 90 years after these accordions were originally marketed. I must watch more of Alf to see if his box uses the Frosini bass system.
Someone is even selling a finto on eBay: Giulietti Accordion Finto Piano Four Row C System Tone Chamber Hand Made | eBay
$3.3k and it's yours.