Solo a Fate?
Hard to do so when I am not allowed to summon my sidekick chocobo due to the queue restrictions.

Kill trash?
Sure. 700 xp per 2 mins of work/cd. That's 21k xp an hour, awesome only 3.76 million to go or rather 179 hours of casual nonsense.

Roulette Dungeon?
15+ minutes of DPS random leveling roulette 60k-300k xp.

Specific Dungeon?
30+ minutes of queue for a 300k+ dungeon.

My suggestion:
Throw some PVE solo dungeon instances into the game, kind of like a tower of sorts >.O (example: Nyzul Isle).
Give perks to people for wanting to do these things, like say...

Materia V of their choice..