Very interesting find! The Faroese ballads certainly attracted the attention of philologists, and it's not inconceivable that Tolkien took at least some interest in them - though it would be nice if we could find some positive indication of this (say, him alluding to them in a lecture or a letter or something). A title like Sigurd the Dragon-Slayer could hardly have failed to intrigue Tolkien, anyway, whether or not he ever actually read it.
Of course, the name Oakenshield has another clear Icelandic source, the name Eikinskjaldi from Vǫluspá. And Tolkien was using Oakenshield well before Smith-Dampier's book came out. Still, it's interesting that the note in TLotR on Thorin's epithet says:
It is said that Thorin's shield was cloven and he cast it away and he hewed off with his axe a branch of an oak and held it in his left hand to ward off the strokes of his foes, or to wield as a club. In this way he got his name. -The Lord of the Rings, Appendix A.III, note
I'd never really thought about it before, but it's not inconceivable that Tolkien was influenced by this ballad (or a similar tradition about an oak club weapon).
It is hard indeed to believe that one of so great wisdom, and of power—for many wonderful things he did among us—could perish, and so much lore be taken from the world.