v1.1
Publishing history:
v1.0: 01/10/24
v1.1: 16/08/25
flathadh f. *[ˈfɫ̪a(h)əɣ] is apparently attested only in the anonymous poem ‘Tha ghaoth ’n Iar cho chaithreamach’: 
A poem in praise of Ailean Dòmhnallach (Allan MacDonald), chief of Clanranald, on his and his brother Raghnall’s departure for the Battle of Sheriffmuir in 1715 (Tobar an Dualchais˄: 109109, Pàirt 1). On p. 121 of NLS’s downloadable copy of An t-Oranaiche (Mac-na-Ceàrdadh 1879), from its Matheson Collection, a pencilled footnote against line 3 (Bha Ailein a’s Rà’ull ann) reads, ‘i.e. Allan MacDonald of Clanranald and his brother Ranald as they left South Uist for Sheriffmuir (Kill. MSS.).’ The reference to the Kill[earnan] MSS (Thomson 1983b, 165; NLS MSS 3781–3784) has not been pursued.
Literally, ‘of goose and of Barnacle goose’.
With (ann) an tathunn ‘in hunting, pursuing’, i.e. ‘being hunted’; cf. SG tathann, tabhann, tafann: ‘bark, barking’ (Dwelly 1911, s.v. tabhann), ‘nagging’ (Wentworth 2003, s.vv. go (on at), nag), ‘urging, pressing’ (McDonald 1972, s.v. tathann), ‘chasing (as hounds), urging’ (Faclan bhon t-Sluagh˄, s.v. tafann), cf. EG tafann ‘hunting (with dogs), pursuing, chasing out, barking (usually of dogs – perhaps the original meaning)’ (≈eDIL˄).
Jakobsen 1928, s.v. flo: [flō].
SND˄, s.v. 1flow: ‘a wet peat-bog, morass, swamp; an arm of the sea, a bight, channel or haven where there is deep water or strong flowing tides’.
An accusative/dative ON fló (a contraction of *flóa) would normally be expected to yield SG *flò. In the context of the poem, flathadh is in the genitive, and it may simply be the genitive of SG flaith 
Dwelly (1911) lists †flaith f. ‘milk’ and †flaith m. in the sense ‘strong ale’ separately.
EG flaith is possibly the same word as EG laith ‘liquor’ (SG laith), but with prothetic f- under the influence of flaith ‘lordship etc.’ or by association with EG finn ‘milk’ (eDIL˄).