Maybe the pearl collier came into the House of Württemberg via the alliance Orange-Nassau x Braunschweig-Wolffenbüttel (on itself a house with many Prussia alliances).
Queen Sophie's grandmother was a sister of the Erbprinz Karl von Braunschweig-Wolffenbüttel, Duke of Braunschweig and Lüneburg. By virtue of this marriage Queen Sophie's grandmother was also a sister-in-law to the Erbprinzessin Louise, born Princess of Orange-Nassau.
Queen Sophie herself would marry Willem III. His mother Anna Pavlovna was a sister to her mother Catharina Pavlovna.
Extremely close and complicated connections. I think that the guess of Martijn Akkerman that the pearl collier returned from Prussia back into the Orange-Nassau collection via Queen Sophie is a plausible one.