Yes, Ice Fang will get written over. Moves that Luxray learns by level up can only be learned only if both parent know it. Ice Fang and Fire Fang are classed as egg moves so they cannot be learned in this way (and you'd need the Houndour to know both anyway).
Egg moves can only be learned if the father knows both moves - they are not passed down from the female. So to get Ice Fang and Fire Fang on a Shinx you need to breed with a male Pokemon that can has both moves.
Lucky for you there are a bunch of Pokemon that can learn both by level up: Arbok, Snubbul, Granbull, Exploud, Hippowdon and Stoutland. (Some of them need a Heart Scale to remember the moves.)