The below video is incredible, IMO. I have never heard this tune before, but I really like it.
I tried to find it over at thesession.org, but the tune associated with the title doesn’t bear much resemblance (if any).
http://www.thesession.org/tunes/display/7
He indicates this is his own rendition - Did he so change the tune that it’s almost a new tune or am I looking in the wrong spot?
That’s not the BCtR that I know. An Old-Time tune, maybe? Sorta sounds like it, if my ears don’t deceive me. Nice one in any case.
This is the one that I know (but it’s definitely an ITM tune):
You can’t go by the titles with these Boney tunes. 
The first is the Old Time tune, listed under that title in The Fiddler’s Fakebook, and also the first entry in The Fiddler’s Companion.
http://www.ibiblio.org/fiddlers/BOH_BONIN.htm#BONAPARTE_CROSSING_THE_RHINE_[1]
In both sources, the ITM setting is listed second, as Bonaparte Crossing the Rocky Mountains. Or was that Hannibal?
The first is the one I know best, since it’s a popular contradance tune (and very nicely done here on mando, I’ll add). But I’ve never heard it in Irish trad. Guess it could be done as a march or polka or barndance. As a barndance, it bears some resemblance to An Comhra Dunn (The Brown Casket).
Then there’s Bonaparte Crossing Himself, Bonaparte Crossing His Wife, Bonaparte Crossing His Legs, Bonaparte Crossing the Intersection…there’s a bazillion of 'em.
Yep, that first video is the oldtime tune by that name. It got popular in the 1960s after being recorded by the Fuzzy Mountain Stringband, from Chapel Hill, NC. I think they got it from the fiddler Henry Reed. Forty-some years later, it’s still very popular around here.
#23 on this page: http://itunes.apple.com/us/album/the-fuzzy-mountain-string/id2476778
And I got it originally from the Fuzzies when I went to school with them in the 70s. They had a lot of tunes from Henry Reed and Tommy Jarrell.
The world continues to get smaller!
The Fuzzies had a tremendous impact around here, and are still a role model for up-and-coming oldtime string bands. Truly one of the under-sung revivalist groups of American trad!
Most of them have moved on, particularly Tommy. (I’m pretty sure he checks in regularly, though). Bill Hicks is still kickin’ it with his incredibly talented partner, Libby. And we’ve got Jim Watson, Alice Gerrard and a bunch of other sometime Fuzzies still making great music here. It’s a very rich and undervalued tradition, IMO.
If anyone else is curious about this stuff, here’s something to get you started: http://www.oldtimeherald.org/
And of course, a shameless plug for my own organization:
http://www.pinecone.org/
Here’s the abc for one version of the tune the OP was talking about. The mando player made some little changes from what I’ve usually heard, but certainly nothing huge.
X:85
T:Bonaparte Crossing the Rhine
T:The Caledonian March
C:aka The Caledonian March 1837
S:Howe’s Complete Preceptor for the Accordeon 1850
M:4/4
L:1/8
Q:120
K:D
FG|"D"A>B AF A2 de|f>e fa "Bm"d2 dc|
"G"B B/c/ dB "D"AFED|"A7"E2 E>F E2 FG|!
"D"A>B AF A2 de|f>e fa "Bm"d2 dc|
"G"B B/c/ dB "D"AF"A7"EF|"D"D2 D>E D2 de||!
"D"f>e fg a2 aA|"G"B>A Bc "D"d2 dA|
"G"B B/c/ dB "D"AFED|"A"E2 E>F "A7"G2 FG|!
"D"A>B AF A2 de|f>e fa "Bm"d2 dc|
"G"B B/c/ dB "D"AF"A7"EF|"D"D2 D>E D2||!
I play the original youtube version, as in the first post. I think many of these old-time American fiddle tunes are great for whistle and flute.
There’s a great version of the tune on John Skelton & Kieran O’Hare’s CD “Double Barrelled”. That’s where I learned it, anyway.