This has been driving me crazy, so I stripped everything down and rebuilt from scratch. I'm not sure why it didn't show up before when I previously cycled through other plugins as the potential cause/conflict, but it appears that Automattic's [WP Super Cache ](https://wordpress.org/plugins/wp-super-cache/) when activated and with caching turned on in the settings admin is what is causing my initial issue.

Turning caching off on this particular plugin (or un-activating it altogether) allows the retrieve button to work properly. My JS skills are woeful and I'm not seeing anything on direct inspection that would cause this problem when it's enabled. If anyone has useful thoughts about the conflict, I'd love to hear them.

It appears that WP Super Cache hasn't updated since October, so it likely wasn't a change on their end, but rather a change when Post Kinds upgraded.