How can I accomplish smooth video playback with QLab 2?
Sounds like a job for either re-compression, solid state drives or both.  It appears that the BlackMagic Television Studio does has the media server that the other Black Magic ATEMs have, so you might also be able to load them in to that depending on their length.

Your bottle neck is almost certainly data rates, and the main options are to increase the rate your system can move data at (solid state drives), reduce the amount of data that needs to be moved (re-compression) or place the data somewhere that it can be accessed more easily (in the media server of the ATEM).