There are some silly, probably useless obstructions, at least in the "honest Fano" case: the Lefschetz hyperplane theorem equate the second Betti and third Betti numbers (resp. Hodge numbers of total weight $2$ and $3$) to the corresponding numbers of the Fano manifold. Fano manifolds of a given dimension are bounded, and explicit examination of the proof should lead to some (probably outrageously large) upper bounds on the Betti numbers. So if the Betti numbers of the CY are larger than those bounds, it cannot be an anticanonical divisor in a Fano.