That kind of "double soundboard" - a removable "false soundboard" (as described by Montal) was quite common in the early 19th century. I haven't heard it described as a double soundboard, but I suppose it could be.
Montal wrote (in his section on piano history) that Pleyel experimented with veneered soundboards in 1830. This was partly an offshoot of Pleyel's attempt to compete with Erard in the area of the harp, as it was initially done for the harp. Here is that passage:
In 1830, Pleyel introduced veneered soundboards in the piano.
This improvement, which astonished the whole world because
it was in opposition to all recognized ideas, produced,
however, happy results.
Dizy, associated with Pleyel for the fabrication of harps,
had been led by various experiments concerning the resistance
of soundboards to glue a thin board of another wood to an ordinary
spruce soundboard, crossing the fibers to give it more
solidity. His friends and workmen tried in vain to dissuade
him from this attempt, which seemed to them to be folly, but
this professor persisted in his idea. The harp thus built did not
truthfully have more power than an ordinary harp, but the
tone gained in the area of quality. Then Pleyel made a trial of
the same sort on a grand piano, veneering a spruce board in
mahogany, crossing the fibers of the woods. The result was the
same; i.e., the tone did not increase in volume, but acquired a
particular quality that was very satisfactory, the treble becoming brilliant and silvery, the middle penetrating and accentuated, and the bass clear and vigorous.
Regards,
Fred Sturm
"Believe those who seek the truth; doubt those who find it." Gide