Apart from I like the idea of feeding my Chocobo, I feel you can think of it as an expense, i.e. the same as the cost of repairing your Armour and Weapons.
So if you still want an alternative idea then you could combining the ideas above and instead of a timer, your Chocobo needs to eat more food after a long battle
If you do the Vath Beastmen Quests. You can buy Food that helps to improve your Chocobo :)
- Mimett Gourd - Enhances Healing
- Pahsana Fruit - Enhances Emnity
- Sylkis Bud - Attack Power Boost
- Tantalplant - Increase Chocobo HP