[Gordis]

Originally Posted by Gothbogg the Ripper

I doubt they would have seen him. Whereas the Ringwraiths were cunning enemies and the cream of Sauron's crop, the Barrow-Wights were merely puppets. Called up by the Witch-King if I am not mistaken.
And whilst the Nazgul were neither dead nor alive, the Barrow-Wights were most certainly dead. So, no ring-vision for them!

I disagree. It is not a question of Power, it is a question of access to the Spirit World.

Putting on the Ring, Frodo is transferred from the ordinary physical world into the Shadow/Spirit world and thus becomes invisible to mortal eyes, to all the eyes of the ordinary physical World. On the contrary, wearing the Ring he becomes fully visible to all the denizens of the Spirit world: ghosts, spirits, wraiths, ringwraiths etc. He would also still be visible to those "who dwell in both words": Calaquendi Elves (like Glorfindel or Galadriel) and Tom Bombadil.

Barrow wights are in essence SPIRITS:

It was at this time that an end came of the Dúnedain of Cardolan, and evil spirits out of Angmar and Rhudaur entered into the deserted mounds and dwelt there.- LOTR, App A.

It is not important that the spirits animated the dead bodies in the Barrows, still all spirits exist primarily in the Spirit World. They would see Frodo with the Ring clearly.

The Nazgûl found one another easily, since they were quickly aware of a companion presence, and could hear the cries over great distances. They could see one another also from far away, even by day when to them a Nazgûl was the one clearly visible thing in a mist. Reader's Companion p. 164

The denizens of the Spirit World had no problems seeing each other, by day or by night. The wielder of the One Ring or a ghost of Dunharrow, or a Barrow-Wight would be as easily detectable to a nazgul as another nazgul: they all share the same World.

It is to see things of the Ordinary physical World (and especially to interact with them) that the nazgul and other Spirit World dwellers needed extra-powers and extra-efforts.