Hello Colin, the Fetherstonhaughs of Carrick have a pedigree in vol. 1 of BLG 1879, which begins with a Thomas ("2nd son of Cuthbert Fetherstonhaugh, and brother of Cuthbert Fetherstonhaugh, Esq. of Dardistown") whose fourth son, Ralph, was created Baronet and was noted to be "ancestor of FETHERSTON, Bart of Ardagh". The arms are given as "same as FETHERSTONHAUGH of Bracklyn"; that pedigree, in the same volume, begins with "Cuthbert Fetherstonhaugh, a scion of the Northumbrian house, was the first of the name who settled in the Co. of Westmeath." His grandsons, via his son Cuthbert, were Cuthbert (of Dardistown, as above), Thomas ("ancestor of FETHERSTONHAUGH, of Carrick", and FETHERSTON, Bart., of Ardagh"), and Francis ("ancestor of FETHERSTON, of Whitrock, co. Longford"). The Fetherstonhaughs of Rockview are also in the volume, the lineage beginning "This is a branch of FETHERSTONHAUGH of Bracklyn. RICHARD STEELE FETHERSTONHAUGH, Esq. of Rockview, co. Westmeath, 2nd son of James Fetherstonhaugh, Esq. of Bracklyn ..."- a check shows James is indeed in that pedigree, a grandson of Cuthbert of Dardistown (aforementioned).
The previous page has Fetherstonhaugh of Hopton Court, with a large block of text (of unknown veracity, sounding very similar to so many "family traditions", but one hesitates to speculate) beginning "The first of the Fetherston family who settled on the Tyne, Northumberland, is stated to have been a chief of those Saxons who landed about the beginning of the 8th century". Presumably this is the "Northumbrian" family to which the Brackley pedigree refers, but Cuthbert's name does not appear (which given the limits of such pedigrees is perhaps understandable). The rest of the passage possibly warrants reading, but is rather too long to include in this already awfully long post -
https://archive.org/stream/genealogicalhera01byuburk#page/556/mode/2up There are two other Fetherstonhaugh families- of The College, Kirkoswald, and an offshoot, of Staffield Hall; neither pedigree makes any reference to those dealt with at greater length above.
Although I have a copy of BFI, and (rather more regularly) use the online version on the Burke's site, I must say I've several times encountered omissions from the Index that fail to account for families that are included in editions of the Landed Gentry.