Because I have one child who had early puberty and no CAH and another child with LOCAH and no early puberty I will tell you what I know.  There is a difference between premature adrenarche and central puberty.  The first sign of central puberty in a girl is breast buds.  If your daughter doesn’t have any breasts, she doesn’t have early puberty.  And if she’s a little chunky it could be hard to tell.  All the other things your describing, oily skin and hair, body odor, are adrenarche.  This is excess hormones coming from the adrenal gland rather than the gonads which is where they start to come from during true puberty.  We all know that our children have a tendency to produce excess male hormones from their adrenals because of the CAH.  Your daughter is probably somehow being undersuppressed, but because of the weight gain, perhaps oversuppressed at the same time.  I think circadian dosing could be just the right thing for her.  She is probably not getting enough meds early in the morning - causing excess hormones (and possibly the good growth too) but getting too much at other times.  How are her labs and bone age?