I'm interested in gravitational anomalies both which arise when coupling to curvature fluctuations or when placing the system on a manifold of non-trivial topology.

Intuitively, we can compare the curvature radius to 1/gap if we have some characteristic velocity. However, an anomaly is some discrete behavior, so even if the curvature radius is much larger than 1/gap, the effect may be hard to measure but it could still be there, so this does not rule out a gravitational anomaly of the first kind.