As a follow-up on Ace-Garageguy's approach, here's a drill cradle I made; it allows hands-free operation of a variable speed drill.
I find the Dremel sanding drum to be an almost-perfect size for mounting the tire. Also, consider taping two fresh #11 Xacto blades together (use a scrap styrene strip to widen the gap, if necessary); that approach will give you a consistent amount of material being removed on all of the tires. Lastly, make the cuts near the edge of the tread pattern, not in the center; mount the edge that was cut on the inside of the chassis area.
For sanding, toss the tire in the freezer for an hour or so. Remove the tire and, working quickly, sand it with 600 grit sandpaper.