Dominion Tower III is a music track that is unlocked when entering any mode at the Dominion Tower (Climber, Endurance, Special, Freestyle, or Spectator).
As entering Spectator or the easy mode of Sunfreet (which does not require other players to be in the tower) does not have any requirements, neither does this track.
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 (entering the arena not required; you can exit immediately via the spiral stairs). This may take several tries due to the random ordering giving you the same track more than once upon re-entering. Alternatively, you can simply stand outside the arena and wait for the track to end and the next would automatically be unlocked and begin.