No, there is no boost for your chocobo - its also quite normal that the bird gets very little exp from just killing mob in the overworld, even if you're at level.

There are two actual good sources for chocobo-exp:
The first is the weekly agenda bonus that you get for killing a certain amount of mobs with your bird by your side - my guess would be that thats what got your main characters chocobo up to a decent level while playing through the MSQ: A couple of weeks of weekly boni.
The second one is placing your bird in a stable at your FC-house (if you have one) or at your apartment-building (if you have an apartment) and feed it with Krakka-roots - other members of your FC can feed him aswell. There is a one hour cooldown on feeding the chocobo and even if you're offline other people in the FC might take care of him!

At the end of the day leveling the chocobo is quite a slow grind with no huge benefits (yes, its nice that your chocobo can heal you, but its not a dealbreaker) and as you said it yourself: you dont need his help right now. With the weekly bonus you'll get there soon enough again.