Thanks again for the clarification. I should have written "when the curvature of metric is degenerate appropriately at higher order"

Thanks for the answer! The coefficients of a Killing field $v$ and its first derivative satisfy a closed system of DE's of the form $dv=\omega v$, and since $\omega$, which determines the rate of change of $v$, is bounded in a neighborhood of the puncture, $v$ must be bounded. I mistakenly thought that there might exist a Killing field with a pole-type singularity at a point, when the metric is degenerate appropriately at higher order.