Frodo's Barrow Dilemma

[Inziladun]

Originally Posted by Gordis

The wights were spirits from Angmar and Rhudaur sent by the Witch-King to inhabit the barrows (see App.A). As spirits they belonged to the Spirit world, to the World of the Unseen, so a person wearing the Ring would be fully visible to them. It was a very BAD plan.

The wights appear to have some notable differences from the Ringwraiths, and there seems to be evidence against wights being able to see one wearing the Ring.

Trembling (Frodo) looked up, in time to see a tall dark figure like a shadow against the stars. It leaned over him. He thought there were two eyes, very cold though lit with a pale light that seemed to come from some remote distance. Then a grip stronger and colder than iron seized him.

FOTR Fog On the Barrow-Downs

Frodo, without wearing the Ring, was able to see the Wight. Compare this to the Ringwraiths, that are completely invisible to mortals unless they wear clothing to give shape to themselves.
If Frodo could see the Wight without the Ring, and it could plainly see him, I am inclined to think it could not have seen him while he wore it.