Dr. Francis and Clement C. Moore

via Find a Grave

Well, I don't suppose Herman Melville would have bumped into old Clement C. Moore at the New York Society Library, on Broadway at Leonard from 1840 to 1856. Moore served as Trustee in 1811-1815 and 1817-1823, so his tenure officially ended twenty five years before Melville first joined in 1848. Melville belonged off and on in 1848 and 1850, before he moved to Pittsfield. Unless maybe they crossed paths while Moore was working on his translation of the Life of George Castriot (published in 1850). 1850 is the year Dr. Moore retired from teaching at General Theological Seminary. Not long after, Moore himself relocated to 444 West 22nd Street between Ninth and Tenth Avenues. He summered in Newport.

One other place Herman Melville possibly could have seen Clement C. Moore in person was at No. 1 Bond Street, the home of their mutual friend Dr. Francis.

From Old New York
via The New York Public Library - Digital Collections

Dr. Francis attended the wedding of Herman's brother Allan Melville and Sophia Thurston on September 22, 1847. Herman Melville, as Augustus Kinsley Gardner reported in 1854, occasionally joined convivial gatherings of literati at the home of Dr. Francis. Hershel Parker observes that

"Melville considered Dr. Francis an intimate friend all through the 1850's, a time when Dr. Francis in public addresses and writing spoke admiringly of Melville and Tuckerman in the same breath."  --Historical Note - Published Poems

Melville's friend and sometime host was Clement C. Moore's very close friend and family physician, too. Henry T. Tuckerman testified to their long connection, and the wonderful, "cosmopolitan" breadth of the Doctor's associations, in his biographical introduction to Francis's Old New York:

But so large was the tolerance and so vivid the humanity of Dr. Francis, that while he was justly considered a thorough representative of the Knickerbocker character,—to whom the New Year's festival, as characteristic of his father's Nuremberg home as of Dutch hospitality, and his old friend Clement Moore's household rhymes of Santa Claus, with the annual schnapps and pipes of this local saint, brought such genial inspiration,—he yet thoroughly enjoyed the thrifty and intellectual example of the Yankee, welcomed the persecuted foreigner, and found delight in the spectacle of variety of race and customs, of excitements and phenomena which transformed, before his eyes, the town, where every one he met was an acquaintance, into the kaleidoscope of modern New York. Indeed, he recognized this expansive destiny in the natural advantages of the island, whose bay and climate Verazzano eulogized centuries ago, and whose noble river Hudson laid open to the Dutch West India Company, whence dates her commercial development. These divers associations only knit the bond of nativity in and identity with New York more strongly. To him, the Jay homestead at Bedford was as endeared because of its Huguenot as the Stuveysant estate for its Dutch traditions; the Tory families of Westchester, the old Episcopal exclusiveness, the wise Rabbis of the Synagogue, and the proscribed Unitarian from Boston, found their due historical and social consideration in his retrospective thought; and the arrival of a Japanese embassy or the Prince of Wales made him a holiday as well as the festivals of St. Nicholas; and loyal as he was to old New York, and proud of Motley's splendid vindication of the Dutch character, he none the less cheerfully grew cosmopolitan with the modern city; and only protested, with a magnetic zeal, alike individual and persuasive, that her Past should not be forgotten amid the absorbing claims of her Present, and the expanding promise of her Future.  --Old New York

Clement C. Moore was one of the pallbearers at the funeral of Lorenzo Da Ponte on August 20, 1838. Attending physician Dr Francis led the train of mourners, as noted in the 1868 article on Lorenzo Da Ponte by Henry T. Tuckerman.