The naming of hymn tunes is arbitrary. Just because it's called "Bunessan" doesn't mean it has any connection with the place at all.

There is no such thing as an ancient Gaelic hymn tune. The Catholic church didn't use hymns in vernacular languages and the Protestant church in Scotland didn't use hymns at all until the 18th century - just psalms, to a very small number of tunes.

The Handbook to the Church Hymnary has this, for the hymn "Child in the Manger":

Mr Lachlan Macbean of Kirkcaldy has kindly supplied the following information andout this hymn and its tune: 'The verses have been chosen from a longer, beautiful hymn by Mary Macdougall, who in the earlier decades of the nineteenth century was a widely recognized poetess in the island of Mull. This hymn she named "The Child of Agh", i.e. of Happiness, Good Fortune, Power, or Wonder.

The tune BUNESSAN was noted down by Alexander Fraser from the singing of a wandering Highland singer. Its bold movements are in keeping with the freedom shown in Gaelic song'. It is printed from Songs and Hymns of the Gael (ed. L. Macbean, 1888).

They don't say that what the wandering singer was singing was a hymn, or that he had any connection with Mull.

Nor do they say what tune Mary Macdougall had in mind, if any.

The words for "Child in the Manger" are unrelated to those of "Morning has Broken", so no link with Mull there either.