So, we ended up having to ask our friends over in the MEAM department for some adequate power supplies to run the drill. Since the H-Bridges from the Lab box only do 2A, Sid suggested that we place multiple H-Bridges in parallel to supply more current.
A 30V power supply and 3 H-Bridges later, our drill circuit was done and mounted behind the PVC monster, allowing us to rotate the threaded rod and translate the backboards back and forth.
Surprisingly enough, at full speed, the backboards have minimal shake. This was our relief since we though we would not be able to fully power the drill, much less rotate the rod with a full load.