Dominion Tower III is a music track that is unlocked when entering any mode at the Dominion Tower (Climber, Endurance, Special, Freestyle, or Spectator).
This track along with Dominion Tower I, Dominion Tower II, Dominion Tower IV, and Dominion Tower V aren't unlocked in a specific order. All 5 can be unlocked by simply entering any mode, leaving and repeating the process. This may take several tries due to the random ordering giving you the same track more than once upon re-entering. The tracks may also be unlocked by spectating fights, which can be done at any combat level.