Welcome, glad you joined us.
I have chamber lots of barrels in lots of calibers. For a .22 I have always simply used a Finishing Reamer. The correct reamer for a .22Short is a .22Short Reamer. A .22LR Reamer may be used but the bullet then jumps a long distance to the throat potentially affecting accuracy. This would normally be a theoretical concern, but for a Gatling Gun - not so much I think.
I would buy a .22LR Finish Reamer and use a depth mic to establish headspace and not look back. This is done by measuring the depth of the rim recess cut by the reamer. If I remember correctly the max to min headspace is about .003\". The nominal cartridge rim thickness is 0.040\" and I have seen the chamber rim depth set at 0.0425\" to 0.045\" and 0.043\" to 0.046\" depending upon the source.
Headspace gauges are handy for checking excessive headspace in closed chambers but since our headspace can be adjusted, just be sure that the chamber is not reamed to deep initially.
gary