What sensitivity of reed switch should I use?
Somewhere around 25 AT will be more than enough to prevent accidental triggering from the earth's field, although you could go higher depending on your magnet and the proximity to the switch.

The trick is to get the alignment right.  You want to align the magnet and reed switch so that the reed switch is magnetized along its length.  See https://standexelectronics.com/wp-content/uploads/Reed-Switch-And-Magnet-Interaction.pdf .

If your magnet is 180 mm from the center (360 mm diameter), the magnet will pass in .0073 seconds, and a reed switch should be fine.  They have operation times on the order of .0005 seconds.

Good luck!