I gather from the wording of your message that it derives from Fr MacMillan's book on migrants to Cape Breton so don't need pointing in that direction. I am sorry that I cannot give you what you are looking for but I can provide some context that may be useful.
I am not sure where the Carinish reference originated but it is the nearest township in North Uist to Ard nan Sruban in the island of Grimsay so the two apparently separate references may well be to one and the same location. The St Michael's, Ardkenneth records of b, m, d do not start until 1829 so there is only one set of records that could throw light on the move of Alexander and family to Benbecula. That is the rent rolls kept by the Factor, Duncan Shaw. As factor, he had the tack of Nunton so individuals were not recorded there. The rolls do identify the tenants in Balivanich but Alexander does not figure. There are two possible reasons. One is that the set of records runs from 1811-1822/3 and then there is a gap until 1835 by which time Alexander had left for CB. He may only have moved to Benbecula after 1823. The second is that the move from North Uist to Benbecula suggests, in all probability, that Alexander was a cottar, an agricultural labourer, herd or similar, rather than having a lease of a croft. In that case, he would not appear in the rolls.
Incidentally, as a result of the much earlier carving out of a separate estate called Belfinlay and then its subsequent re-integration into Nunton, there was an overlap where a triangle of land could have been called either or both of Nunton and Balivanich so that might just indicate where the family was based. The Beatons emigrated from another Benbecula township, Hacklet. I hope this helps a bit.