So if im reading your question right(and if i have my facts straight), you can breed your rank 6 chocobo a maximum of 10 times(this is the max for all birds regardless of rank). I dont really have any breeding strategies to share however, as i stopped around rank 4 and havent gotten back to racing in ages. Its on my to do list. Im sure someone else can give you a better response on that front. Good luck!