I brooded my first birds in my garage, used 18" of wood shavings and it worked well. When I started to smell bird stench, I'd add more litter. That said, to LG's point... I only did a few in the garage so dander wasn't too bad. But when I put them in their winter housing and when I brooded lots of birds in my barn chicken dander was virtually everywhere. There's no way I could have tolerated that in my garage.
A 4'X4'X4' chicken tractor with 1/2" welded wire in your run Might be your best option, but I'd also add a few 3-4 hot wires around the base of your run or use a 2' skirt around the outside of the tractor to deter digging.