Since you asked about primary resources, you might want to look at the personal papers of the President of the BUS at the time of the 1837 Panic: Nicholas Biddle.

His papers are apparently in a collection at the Library of Congress which has excellent microfilm equipment which should make looking through the papers relatively easy.

You might also want to ask the specialists there if the collection is complete or if some of his papers are split up among several archives.

Biddle is originally from Pennsylvania, from a family closely linked to the East Coast commercial elites (particularly those from his hometown). At least one of his brothers (that I recall) was a congressman at one point and others went on to have other major roles in American political and military life. His brother James Biddle was one of the premier naval figures of the period, even achieving the honorary title of Commodore (head of a squadron) which was the highest honest you could have during the early years of the US Navy.

With this in mind you might want to analyze to what extent Biddle's policy was influenced by the interest of traditional commercial elites. I study political and diplomatic history, not economic, but from my reading of Daniel Walker Howe, Edward Baptist, and other serious scholars who have spent time analyzing aspects of banking policy at the time, the idea that the banks were tools of more traditional elites was present among those on the Western frontier.

Because Biddle is from Pennsylvania itself, you might also want to contact the Pennsylvania Historical Society. They're great, specialized, very serious, and according to their website they seem to have some assorted papers of Biddle from several periods (though they all seem to be prior to the 1830s).

Hope it helps!