So on the Daily Realm, I encountered the Pumking once, didn’t get a chance to extract him (he was a resurrect of another creature, so extracting got me a core of the original creature).

So… How do I get Pumking cores? Thanks in advance!

You wait until Zack send out the update that lets it spawn normally. Originally it was going to be part of a yearly Halloween event, but that kinda got nixed when he moved on to Siralim 2.