The piece requested by Beauchamp in 1826 was almost certainly the undated

"BONAPARTE'S RETREAT!!! From Moscow: Composed by Alexander Campbell. Published by John Cole, Baltimore."

https://levysheetmusic.mse.jhu.edu/collection/040/028

Along with "Grand Waltz, by Mozart," "Believe Me, If All Those Endearing Young Charms," "Marseilles Hymn," etc., the title appears in program of classical and semi-classical pieces advertised in the Baltimore American and Commercial Daily Advertiser (Aug. 23, 1831), p.3.

"Bonaparte's Retreat from Moscow" is mentioned in passing as a familiar piano composition in the Charlotte (N.C.) Journal (July 21, 1842), p. 3; The (Boon's Lick, Mo.) Times (Sept. 5, 1846), p. 2 (as "Bonaparte's Retreat," in a program of violin and piano music); and the Galaxy Magazine (Oct., 1866), p. 281 (mentioned as a piano piece along with "Money Musk," "Fisher's Hornpipe," and "Java March").

In folky vein, The Old Abe Eagle (White Oaks, N.M.) (May 24, 1894), p. 2, reports (in fiction) that,

"Hank...handled the bow dexterously and often relieved the monotony of cow boy [sic] life by drawing 'The Arkansaw [sic] Traveller,' 'Fisher's Hornpipe,' 'Bonaparte's Retreat,' or some lively jig or cotillion from [a] battered old violin."

Several bars of Campbell's composition (divided into "Slow March" and "Quick March") are reminiscent of the fiddle tune (or vice versa). Many fiddlers play the piece faster and faster. Coincidence?

The disastrous retreat took place in 1812. I've been unable to ascertain anything at all about Alexander Campbell. Was he the celebrated theologian of that name?