That case that alleged the Alexander Backer Company recast existing pieces? I think I found the source of the recasting, but it might actually be the correct item for the prop as well.
19th Century Wedgwood Black Basalt Bust of William Shakespeare
"Marked "WEDGWOOD" "ONS" Victorian date letters for 1864 - 12 3/4" high"
Seems to be part of a pair that included John Milton:
"This pair of basalt busts are in the form of John Milton and William Shakespeare. They are marked Wedgwood and date from the 19th Century. They both measure approximately 12 + 7/8 inches tall 8 + 1/8 inches at the widest point."
When it was recast, an entablature was added of Shakespeare's name:
So, the question is...which version was used in Picard's quarters? I can't quite tell from the screencaps whether there is an entablature on the prop, or not. The top ridge of the pedestal base could be what is seen in the image, or it could be the name plaque. Can anyone tell for sure?