After exhausting all other options, I uploaded the plugin to another site I manage, and it worked perfectly. Once I narrowed it down to an issue with a specific domain, I was able to identify the problem in the active theme.
The original theme I built for the site in question had some YouTube functionality baked in (major no-no), and that’s proved to be the source of the conflict. Conveniently, that theme is being phased out, so this issue will resolve itself once the transition is complete.