SO not sure if this is how it supposed to work or not, but the pet is auto stabled when you are OUTSIDE the dungeon.. this only happens when that pet is a mount.. as soon as you pop to the other side, its stabled.. and you cannot use the stable pole.. it just says that your stable is full.. A pet that is not a mount, cannot leave the dungeon either, so it is stuck in the dungeon, until you logout or call it with a pet summoning ball..