With my supposition that Dupuy's royal bust depicted Louis XV, it is likely that it was the work of sculptor Antoine Coysevox (1640-1720). Coysevox's bust of the king, sculpted circa 1719 when he was nine years of age, is the best candidate for a number of reasons, including Dupuy's known relationship with the sculptor. Among the items brought to Canada by the intendant was a two-foot tall white marble medallion. The medallion's gilt border framed an inscription that read "par Ant. Coysevox, pour marque singulière de son amitié" (by Ant. Coyesvox, as a singular mark of his friendship).
I have yet to find a bust of Louis XV any earlier than Coyesvox's 1710s model. As the king only ascended the throne in 1715 at the age of five, and in light of Dupuy's friendship with the sculptor (who died in 1720), I think that this conjecture is plausible. A surviving example of the Coysevox bust can be seen in the collection at Versailles, and the Louvre owns a model in terra cotta.
I haven't found anything matching this description in a modern-day collection, but a pair of circa 1745 busts of Louis XV and his Polish-born queen, Marie Leczinska, each showcase a lion at their base. These busts are made of tin-glazed earthenware (faïence) and are based on models prepared by Jean-Baptiste II Lemoyne (1704-1778).
Dupuy's bust must have been an impressive sight! In addition to being visually striking, such an object would have helped to underscore the power of the intendant, sent to Canada by royal appointment, over judicial and economic affairs in the colony. Although a bourgeois, by all accounts Dupuy used furniture and decorative objects to set an aristocratic tone worthy of Versailles in ways that mirrored his non-noble peers in France, to say nothing of actual noblemen and women. The rise of the Parisian bourgeoisie had its origins at the turn of the eighteenth century, and bourgeois financial ventures, combined with administrative service, made for a society that was beginning to accommodate mobility within the hierarchical structure of the Ancien régime.
Below is not the best quality image, but one that shows the regent for Louis XV, his cousin Philippe d'Orléans, with his son placed in front of two busts. The bust occupying the place of honor over the fireplace is that of Louis XV. A bust of Louis XIV can be seen just to right on a small console. The primacy of Louis XV's bust is clear, and the one depicted in this painting by an unknown French artist is almost surely a model by Coysevox.