Kingdom Stables Collar (teleport to San d'Oria chocobo stables | 10 charges)
Republic Stables Medal (teleport to Bastok chocobo stables | 10 charges)
Federation Stables Scarf (teleport to Windurst chocobo stables | 10 charges)
For the Pyracmon part of the event, to get the nation teleport collars, the Eerie Cloak, and the Eerie Cloak +1:
Speak to the Moogle outside a city. He'll send you to kill Wraith Bats first.
Kill up to 30 Wraith Bats (more is overkill), then RETURN TO THE MOOGLE for the "enlightenment" effect so that you are able to do damage to Pyracmon.
Tell the Moogle that you're ready to take on Pyracmon.
Each Wraith Bat you killed, up to 30, makes the "enlighten" effect you get from the Moogle let you do one point of damage (per bat killed) to Pyracmon when you go after him. So 30 Wraith Bats killed = 30 points of damage you do to Pyracmon per hit.
If you don't kill more than maybe ten or eleven bats after you've received the Eerie Cloak and are trying for the +1, you'll get food like Jack-o'-Lanterns and Pumpkin Cake.
If you're patient and kill the full 30 bats or a little less, you can easily solo Pyracmon numerous times to get the three nation teleport pieces that drop from him, especially if you get Regen from the nearby Field Manual or by any other means. You can sometimes get two or more to drop in one kill.
If you do want to solo him, keep in mind that he works like the mobs in Campaign where mobs can't be "claimed." Anyone can jump in.
Oh, and when he readies Monacular Scowl, unlock and turn your back to him, because if he takes off your enlighten effect there's no point in continuing to fight him as you'll do no damage at all.