Your Chocobo gains experience based on your current experience cut in half. So if you fight a mob that gives 1k Exp. Your Chocobo is going to get 500. Your Chocobo levels as fast as you want them to level. Your best bet for leveling your Chocobo is a chain party, abuse the Chain Bonus. Getting up to about CB of 5 or higher will net you some yummy experience for you companion. I hope this helps. My officer Darth Bane tested this in our server on multiple mobs. He would fight a mob his level, check the experience, pull his companion out, fight the same mob. Watch his experience get cut in half. So my overall suggestion is find a FATE party and don't worry about the actual reward from the FATE itself, but go for the multiple-mob chain bonuses. Your Chocobo should level rather quickly from it.
^This is about your only option.
I personally think that there are 2 big problems with the companion system:
1) We can't use another mount while the companion is outside
2) We can't use the duty finder either.
Until those two are addressed I don't think there will be a lot of situations when I want to take out my chocobo other than completing a high level entry of the hunting log.