Just some ideas that popped into my head today while I was fooling around, most jobs have some kind of stance that alters their play style, so why not give something similar to DRG?
Ability Name: Spirit Scout (Stance 1):
Flavor Text: Sends your Wyvern high into the sky to keep a watch out for the Dragoon.
Effects: While under this stance the Wyvern disappears from combat, not being able to be attacked from any damage nor can it attack any target. Dragoon gains the ability of Wide Scan III, gains an increase to defensive stats: Vitality+10, Agility+5, Defense+45, Magical Defense+15, Enemy Critical Hit Rate-15%, Parry+15 and Evasion+15. CANNOT use Breath Commands while under this stance.
Duration: 2 hours
Recast: 3 minutes
Ability Name:Spirit Bond (Stance 2)
Flavor Text: The Wyvern and Dragoon share their strengths and weaknesses.
Effects: Grants the Dragoon an increase to offensive stats: Strength+10, Dexterity+5, Attack+25, Accuracy+25, Critical Hit Damage+10% while decreasing their defensive stats: Vitality-10, Agility-10, Defense-25, Magical Defense-15, Enemy Critical Hit Damage+10%. Grants the Wyvern an increase to offensive stats: Attack+30, Accuracy+30, Critical Hit Rate+10%, Haste+15%.
Duration: 2 hours
Recast: 3 minutes
Ability Name: Spirit Breath (Stance 3)
Flavor Text: The Dragoon focuses his strengths within the Wyvern to increase survivability.
Effects: Grants the Wyvern with various beneficial stat increases: Wyvern HP+500, Healing Breath Potency+5%, Wyvern Defense+30, Wyvern Magical Defense+30, Elemental Breath Potency+10%, Wyvern Damage Taken-30, Breath Recast-10. Grants the Dragoon with various stat alterations: HP-250, Defense-15, Magical Defense-15, Attack Speed-5%, Vitality-15.
Duration: 2 hours
Recast: 3 minutes
Of course only one of them can be active at a time newer applications overriding previous stances. While under all these stances the Two Hour ability "Spirit Surge" will still be available since the Wyvern is still technically "called out". Also just to restate, while under the first stance the Wyvern will NOT be able to use Breaths of any kind.