Seabreeze, I'm not familiar with your Cambridge Audio CD player, but, considering that it could be your no-name IC that's making your CD player sound bright, not the player itself, you may want to audition what is generally regarded as a neutral cable, as well. A warm cable could, potentially, push you too far in the opposite direction.
You never mentioned a price range for the IC you want to buy, information that would be helpful.
You never mentioned a price range for the IC you want to buy, information that would be helpful.