There's no chocobo boost.
If you only intend to play that one job, then you're probably out of luck. You'll also be level 80 by the end of the HW content if you keep doing daily roulettes as well.
So you'll probably end up leveling a second job anyway if you want playing that character to be at all interesting.
Now I get that you don't want to level from scratch, so you have 4 other options available; Red Mage, Samurai, Gunbreaker and Dancer.
Pick RDM or SAM and you'll be progressing through the story at a relative level, and getting the optimal EXP from fighting in the overworld, thus levelling your chocobo.
Your best bet though, is to stable it and feed it every day. It'll still be slow, and you'll need to take him out every now and then to actually rank up, but it's better than nothing.
Or, as it's an alt, don't even bother with the chocobo.