I asked a few people including Bryan himself and nobody could come up with a name. Bryan kindly told me that he got the tune from Paddy Cronin and that Paddy was well known for picking unusual tunes from books.
After a bit of research and the great help of "sebastian the m̈etapop", it turned out that this tune was a version of the well known tune "A Fig For A Kiss". Here is what’s in the Fiddler’s Companion :
FIG/FEG FOR A KISS , A. AKA and see "Two in a Gig." Scottish, English; Jig (9/8 time). E Minor. Standard tuning. AAB. A distanced variant of “Fig… ." John Glen (1891) states the earliest appearance of the tune in print is in Robert Bremner’s 1757 collection (appears as “A Feg for a Kiss”). Bremner’s version is identical to that printed by Preston. Kennedy gives the alternate title “Two in a Gig.” Source for notated version: copied from Preston’s Selections of Reels and Country Dances (London, 1768) [O’Neill]. Bremner (Scots Reels), 1757; pg. 74. Kennedy (Fiddler’s Tune-Book: Slip Jigs and Waltzes), 1999; No. 21, pg. 6. O’Neill (Waifs and Strays of Gaelic Melody), 1922; No. 200.
T:Fig for a Kiss, A
S:Preston’s Selections of Reels & Country Dances, London 1768
E2 B BGE BGE | FAE AFd AFD | EFG ABc BAG | BeB AGF E3 :||
eBe geg bge | dAd fdf afd | eBe geg bge | gfe Be^d e3 |
eBe geg bge | dAd fdf afd | e/2f/2gf edc BAG | BeB AGF E3 ||