The Kerr initial data (in EinsteinInitialData/IDAnalyticBH/src/Kerr.c) has a defect along the z-axis (that is, the spin axis of the hole). The three-metric (for instance) is not smooth across this axis, leading to discontinuities in second spatial derivatives near there.
This appears to be due to overly crude safety measures protecting from division by zero when calculating this data. Tweaking these measures for more self-consistency leads to much smoother behaviour. I'm attaching a simple patch that fixes the problem at least for metric_type = physical.