Use a really thin (4mm thick by 20mm wide) flat chisel to slice the tip edge. It works better than knife. Take small cut, 1mm to 2mm rotation when cutting the edge.
Keep the cue in the case. Heat will absorb internally by the case more slowly. With the cardbox and cuecase, the cue should be safe.
If you got a triangular tube card box, that would even be better than the square tube card box.