I think that becoming invisible from the Ring is a sign of weakness. Sauron never became invisible because he could master the power of the Ring. I belive that bombadil is too powerful a being to be controlled by any object.
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Ash nazg durbatuluk, Ash nazg gimbatul, Ash nazg thrakatuluk, Agh burzum-ishi krimpatul (One ring to rule them all, one ring to find them, one ring to bring them all and in the darkness bind them)
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