Are you already taking a multi for your total health? When I took a hair multi, it made no difference to me, so I opted to take a full multi which is what I had been doing anyway all along and no problems here.
But if you really need a hair vitamin, just make sure your body is getting enough of the nutrients it needs and that you don't OD in the oil soluble ones by adding a hair one. If your body is not getting what it needs, the whatever you take for your hair will be hogged by your body and so do very little for your hair.
My other question is, does Biotin break you out even when it is part of a multi, or when you take it alone? I believe people get breakouts when they take biotin on its own in very high amounts, or when it isn't part of a complex. Some complexes have very small amounts 400 mcg while others have 3000 mcg. Are you sure the low dosage ones break you out too?
Anyway, to answer you question,
here's a hair, skin and nail multi w/o biotin but I don't know if it is "the best". Otherwise consider a hair multi that has low biotin dosage.