Byakko was added in patch 4.2 of the Stormblood expansion.
Byakko becomes available at level 70 as part of the “An Auspicious Encounter” quest, which is given by Soroban – The Ruby Sea (X:5, Y:15).
The trial has a minimum ilevel of 325 to enter and rewards you with:
Allagan Tomestone of Poetics: 12