Fal Agh Bagh means Power and ......
The other option I found was from gaelic prose and involved something to do with 'authority of the bosom'.
Authority of the bosom is never to be trifled with, and nobody I know would dare name a creature even as noble as a racehorse in connection with it.
There has been intense research into the derivation of the name Fal Agh Bagh by some eminent scholars (scholars are always eminent, lawyers only sometimes) of matters equine and Gaelic in both the island of Ireland and continental Europe.
The best theory any of us can come up with is that Fal Agh Bagh is not merely an anglicised, but also a bowdlerised, rendition of the Gaelic phrase 'fag an bealach', which means 'clear the way' or, if you prefer, 'get out of the way'.
The motto of the Royal Irish Fusiliers, once a regiment of the British army, was Faugh A Ballagh, which is a reasonable rendition in English phonetics of the Gaelic original. The curator of the Ulster Tower at Thiepval, where the Battle of the Somme was fought, confirmed this to me when I visited there a couple of weeks ago. He also casually observed that if pronounced using the hard Ulster g sound, however, this motto sounds perilously close to describing the male organ and its overall purpose (actually, he put it in less delicate terms than that).
Off the cuff though his observation may have been, it nevertheless seems pertinent. One can imagine the unfortunate lady guests, including perhaps even Queen Victoria herself, blushing at first and then wincing as ever more enthusiastic toasts are raised at regimental dinners and cries of Faugh A Ballagh are uttered by the predominantly youthful male attendance. It might therefore be the case that Fal Agh Bagh evolved as a less troubling phrase to anglophone ears.
I'm sure you are all greatly relieved to see this pressing question receiving the attention it deserves.