There is no way to have the Chocobo use skills manually, but you can stick him in Healer stance to make him prioritize it.

Choco Regen is a weak heal over time, so it is likely that the Chocobo just won't use it if it is already on and it is a weak Regen effect as it is. If you keep leveling up your chocobo in the healing role, however, he will get direct heal spells that are more potent and he will use as needed as they don't have a status effect attached to them.