I was able to use a currently running Japanese Mystery Gift code event to get a Beast Ball prior to the post-game.
It also took longer than I expected to get a Modest or Timid nature on Eternatus, but I ended up getting Timid on back to back attempts (the first one had poor IVs on the non-31 stats). I eventually settled on keeping one that had 19/31/21/31/31/31 IVs, as getting 5 IVs would already be a huge challenge without the ability to control the nature with Synchronize.
OK this sounds more of a waste of time than anything I ever did But congrats