When Pinata central asks for a pinata to be sent, accept the challenge and send the pinata.
If you don't have access to the pinata being ask for, you can decline and it will not effect your achievement chances.
If they ask for a valuable pinata, it doesn't matter which you send, there is no quota to meet, you could send a whirlm for all it matters.
If it asks for a pinata that you do not have in the garden, but have had previously, You should have enough time to get the hunter to fetch you one before the time expires, so accept it and just place the box. I'm pretty sure I had her fetch two pinatas before time expired before, so no worries.