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A quick check of the logbook holdings pages of the NARA seems to indicate that they don't hold any copies of the Pickering's logbook:
There is also this article by Dr. William Thiesen: Benjamin Hiller & the Cutter Pickering in the Quasi-War with France (and the 2015 Coast Guard blog entry which appears to be by the same author and has similar information: Honoring History: Revenue Cutter Pickering « Coast Guard Compass)
That article offers a pointer to the 1930s work Naval Documents related to the Quasi-War between the United States and France which is available online at ibiblio.org. There is mention on PDF page 319 (document page 295) in volume 4 of the encounter, but unfortunately it seems to be only a brief entry.
You might try contacting Dr. Thiesen to see what his sources were, but this quote from the article probably sums up the situation:
The case of Pickering underscores the obstacles to documenting the early history of the Revenue Cutter Service and of any ship that put to sea never to return. When Pickering went down in that September storm, it took with it many documents, including bookkeeping records, letters, logbooks, and memoirs. In addition, the sacking of Washington by the British in the War of 1812 and fires at the Treasury Department in the early 1800s destroyed much of the archival material that was left to document the story of this historic cutter and crew. Today, only contemporary accounts from newspapers, a few Treasury letters, and some naval records can trace Pickering’s distinguished history.