The system is separable after you measure out your ancilla. Imagine a stream of qubits B_1, B_2, ... prepared in +X who interact weakly with qubit A prepared in +X. and are then strongly measured projectively. After the brief interaction, A and a certain B will be entangled. Projectively measuring this B updates our knowledge of A, but due to the weak entanglement, the adjustment is small and incremental. The state of such B qubit is known and thus there is no remaining entanglement between A and B, and thus A and B are separable. After many such Bs, an outside observer privy to all measurement outcomes will understand that A is doing an absorbing random walk to the poles of the Bloch sphere. Throwing away this information and taking an ensemble average, you'd see an exponential decay of coherence.
This is a toy model of a qubit being weakly measured by an incident flying field.
> Throwing away this information and taking an ensemble average, you'd see an exponential decay of coherence.
That is described by a density matrix but is not a pure state. The true state may be a pure state (because as you said the systems are separable after the measurement) but our description is a mixture reflecting our imperfect information (and not a superposition of the pure states corresponding to the potential outcomes).
This is a toy model of a qubit being weakly measured by an incident flying field.