So if the Russian American Company library in Sitka, Alaska had the same number of books - about 1200 to 1300? - from 1803 to 1867 it would have been the largest library for thousands of miles around for many years.
The Library Company of Philadelphia's library, about 2,881 miles from Sitka, acquired the Loganian Library of 2,600 books in 1792, and thus was already larger than the Sitka Library in 1803.
Washington DC is only 2,871 miles from Sitka. The Library of Congress might have already been larger than the Sitka library when it was burned in 1814 by the British. The Library of Congress began to be reassembled in 1815, and Thomas Jefferson's entire personal collection of 6,487 books was purchased.
And Jefferson's home in Montecello was only 2,854 miles from Sitka, Alaska, so he was likely to have had a larger library than the Sitka one a Montecello for some of the years after 1803, though he may have kept some or all of the books at the White House in 1801-1809.
New Orleans, Louisiana, is 2,861 miles from Sitka, Alaska, and it's population grew from 8,500 in 1805 to 102,000 in 1840. The first college in New Orleans was founded in 1834, and I am sure that there were a few libraries with over 1,200 books in New Orleans by 1867.
What is now Moravian College, Bethlehem, PA, 2,841 miles from Sitka, was founded as a female seminary in 1742. It is possible that it might have had a library of more than 1,200 books by 1803.
Lancaster, Pennsylvania, is 2,837 miles from Sitka, and had a rapidly growing population of thousands by 1801. The first college in Lancaster was founded in 1787. Lancaster was the capital of Pennsylvania from 1799 to 1812. Thus it should have had one or more libraries with at least 1,200 books by 1867.
Pittsburgh, Pa is 2,676 miles from Sitka. Pittsburgh Academy, founded in 1787, became the Western University of Pennsylvania in 1819 and might possibly have had a larger library than Sitka during that period.
Lexington, Kentucky is 2,626 miles from Sitka. Transylvania University was founded in 1780 and moved to Lexington in 1789. It might possibly have had a larger library than Sitka during the period in question.
St. Louis, Missouri, is only 2,386 miles from Sitka, Alaska. The population of St. Louis grew from 1,200 in 1804 to 77,870 in 1850. The first college in St. Louis was founded in 1818. It seems certain there were a few libraries with more than 1,200 books in St. Louis by 1867.
Santa Fe, New Mexico, is only 2,004 miles from Sitka, Alaska and was founded in 1610. But I rather doubt that it would have had a bigger library than Sitka by 1867.
San Francisco, California is only 1,457 miles from Sitka and rapidly expanded into a major city during and after the Gold Rush. Thus it may have had libraries with more than 1,200 books before 1867.
Thus, during the period of 1803 to 1867, the location of the nearest library with more books than the Sitka library probably changed several times as cities closer to Sitka grew and libraries were established closer and closer to Sitka.