I agree with both statements. But from them I infer 3 conclusions:
1. If the theory is not deterministic, while the evolution of the state is deterministic, then there is something in theory that is not uniquely defined by the state evolution. We don't know what this something is, but since it must exist (otherwise we have a logical inconsistency) let us give it the name ##\lambda##.
2. The only role of the Hamiltonian in the theory is to govern the state evolution. This state evolution is local because the Hamiltonian is local. But since ##\lambda## is not uniquely defined by the state evolution, it follows that the evolution of ##\lambda## is not uniquely determined by the Hamiltonian. So the fact that the Hamiltonian is local does not imply that the evolution of ##\lambda## must also be local. In other words, non-locality of ##\lambda## is not incompatible with the quantum theory.
3. Just by 2. we cannot decide whether the evolution of ##\lambda## is local or nonlocal, both options are open. But if some additional properties of ##\lambda## are assumed (determinism is not one of those assumptions), then Bell theorem proves that the evolution of ##\lambda## (deterministic or not) must be non-local.