I just noticed that the 1 hour cooldown for Chocobo training while stabled is related to the character, not the chocobo.

I have a solo FC and just made an alt character. When I have both my main and alt birds stabled, I can train them both with 1 character, then log into my other character and immediately train them again. You can do that every hour.

In theory, you could have a stable of 16 chocobos and have all 16 FC members train all of them every hour.

Not sure if this is a bug or not but certainly speeds up the otherwise horribly slow chocobo leveling process.