What follows is an in-game explanation with a heavy dose of speculation on my part. However, most of that speculation is not just mine… In short, you cannot permanently destroy a wraith created using any Rings of Power as long as the One Ring exists.
The Barrow Wights are undead wraiths/spirits called forth by the Witch King of Angmar. We know that the first of the Nazgul was a powerful sorcerer as was show by him casting spells (like the ones to break the games of Minas Tirith) and breaking Frodo's sword with but a gesture. Now, it is no stretch to imagine that Sauron might have passed some necromantic knowledge to his servitor. In addition, assume that the Witch King was using his ring of power to extend his abilities. Thus, the barrow wights were created using, in part, the power of the rings. Thus, nothing can break those spells until the One Ring gets destroyed. At least, that is how I see things. You might disagree and that is fine since there is no clear explanation from Tolkien.
However, you can dispel a wraith for a time: just like the Nazgul were. Eventually, the wraith will re-form in a weaker state and start growing in power. The more "forcefully" the wraith is destroyed, the longer before it can re-shape and weaker it is once it does so.
A note on Tom Bombadil: He is a wild card in the books, a nod to many different things not really explained (source) and might even be Ilúvatar. As such, anything that Tom does needs to be taken with a grain of salt.