The 2-Headed Dragon has 1,000,000 HP. His abilities make him more of an annoying opponent than a difficult one, as the player will mostly have to be careful to navigate through the Boss' trickeries to succeed.
The Dragon attacks up to three times per turn. Its attacks involve Headrush, for physical damage and Paralysis to a single unit. Magic Bite for physical damage and MAG/SPR Break and Silence to one unit. Both status ailments last one turn. Double Break Breath inflicts Fire and Ice magic damage as well as ATK/DEF 50% Break to all units. It will use only one skill between Headrush, Magic Bite and Double Break Breath and only once per turn.
Its conditional attacks involve, Swallow, which is an instant KO to one unit (if Magic Damage is inflicted on an even turn, it will be used four times), if Special Ability damage is inflicted on any turn, it will use Dragon's Voice which breaks the party's stats by 100% for 999 turns, rendering them absolutely weak.
At thresholds 80%, 60%, 40% and 20%, it will use Boost, which charges for Release, which deals major magic damage and blind and silence to all units. If Swallow and Release are used on the same turn, Release takes precedence, preventing Swallow.
The Dragon resists all elements by 50%, except Ice (0%) and Dark (80%), is immune to Breaks and immune to all status ailments.
As the player can notice, the Trial's difficulty weights on the players inability to damage the dragon through the usual means, namely the units skillset. That is not to say that abilities cannot be used, but rather the player has to be very careful of the abilities it can use and when. Magic Damage can be used on odd turns (now made easier to keep track on due to recent updates). Given the limitations on the use of abilities and the Boss' resistances, it is rather unwise to attempt to inflict some kind of debuff (Imperils being the only ones that can affect it).
As such the main means of damaging the Boss are normal attacks. Although this may look obnoxious, the Dragon's DEF is pitiful so it compensates for being otherwise unable to damage through abilities. Units with Dual Wield-like abilities (innate or not) will be valuable for this. Equipment like Aurora Scarf or Twin Lance (which provide Two-Hit Combo) will allow units to attack twice with their weapons, which also contributes. If combined with Dual Wield it equals to four attacks. Limit Bursts are also valid to damage the Boss.
Supportive abilities like Buffs or Healing abilities can be used without penalty, and are encouraged. Luka and Fina's healing abilities are useful since they heal for decent amounts while remove the threat of Dragon's status ailments. Although both units will need to be maximized.
Ability Counters like Winter Palisade can trigger Dragon's Voice, although normal Counter is safe (since it retaliates with a normal attack rather than an ability).
There are ways to navigate through Dragon's Voice however. If the player has a unit that can prevent stat breaks, they will be able to nullify the threat of Dragon's Voice and damage the Dragon through Special Abilities (Magic Damage is not exempt from the Swallow rule however). Veritas of the Earth is extremely useful for this as he can prevent Stat Breaks, is a Counter Tank and can provide Damage Mitigation, all of which are incredibly helpful.
On the turns it uses Release, Guarding is a good idea to keep the damage at bay. Swallow can be provoked and with Instant KO nullifying equipment will also reduce the Dragon's threat.
Dragon Killer and related abilities will be extremely helpful. They can be found on the Esper Odin, through farming TMR (found on a Friend Point Unit), or otherwise found on several pieces of equipment (mostly TMR or event equips), so there is plenty of room to deal great damage to the Dragon.
The Mission rewards are insignificant, so the player is not required to abide by them to gain prizes.