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.)