About Gandalf seeing through the ring-veil (I know, technically not correct)... I understand why Tolkien kept out of it - it is hard to imagine him not being able to, being tricked by such a toy feature. Otherwise clearly assigning more power to Gandalf against the ring is not neccessarily good - how to explain that the guy doesn't want to give it a try to destroy or even use the thing?
That being said for the narrative choice, he probably could, probably even the higher elves could.
We shouldn't forget that the power Bombadil had over the ring, wasn't just the ability to see someone wearing it, but not to let the ring lead him to the other side, not disappearing from the land of living while wearing the piece.
Now back to the original question. To answer it, we should first know how the Wight the hobbits "saw". All the groping and the eyes lit from beyond probably mean a sight from the shadow world, feeling and seeing just enough from the world of living to take its victims. So probably it was a bad idea to put the ring on in order to become invisible to the barrow creature. He'd more like had gotten a view of the Frodo Bagging in his best shine and brightness.
Which on the other hand, might not have been bad for the initial plan (save your skin leaving the friends behind). The B-W was powerful alluring fresh meat, that is foolish enough to fall asleep in his range, but being affronted by a power like that of the ring, would have probably taken him aback. I guess he would think that he has made a mistake capturing something he is not entitled to and step back.
Of course that would have still meant saving himself in the moment and letting his friends to the will of the BW. I think this episode was one of the most important part in the preparation of Frodo as a Ringbearer.
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"Hobbits! Well, what next? I have heard of strange doings in this land, but I have seldom heard of a hobbit sleeping out of doors under a tree. Three of them! There's something mighty queer behind this."
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