After scoring the router bit shaft with a vise on one occasion and cutting my finger on another, I came up with this jig to safely hold the router bit while loosening the bearing screw. The jig can be held in your hand or placed in a vise. To use it, retract the plunger, insert the bit in the hole and release the plunger to grip the cutters while you loosen the screw.
I cut a block of wood to the size shown in the drawing below. Drill holes for the bit shaft, dowel pin, and the plunger mechanism.
The dowel pin on the opposite end provides additional support for larger, profile bits.