When used on a chocobo the dye will last indefinitely as long as your chocobo is not re registered. Recommended to only use after your chocobo has been retired and all of its stats locked in
Since you gain 5 for a first place, 3 for a second, 1 for 3~7 and you cant guarantee a win even with a maxed out chocobo, it takes a fairly while to get them.