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Oh, he is in the White Council alright. Galadriel herself has called him the wisest Elf in Middle-Earth.
That was to keep their family relationship on an even keel. What she really thought remains unpublished, though she may have whispered it to Gimli after his arrival in Valinor. Oh, he is in the White Council alright. Galadriel herself has called him the wisest Elf in Middle-Earth.

Celeborn undoubtedly was included as one of the "and other lords of the Eldar" of the passage in The Silmarillion in which it named the five major members of the White Council and told of Galadriel wanting Gandalf to be their leader in lieu of Saruman who was ultimately chosen for that position. This is all found in the last chapter called 'Of the Rings of Power and the Third Age'. (On page 360 in my paperback copy with the picture of Feanor and the three Silmarils on the cover.)