Hi Kate,
Thanks for reaching out. Your system information shows your site’s running version 4.2.7 of The Events Calendar and version 4.2.6 of Events Calendar Pro—the [tribe_event_inline] shortcode is a feature new to version 4.4 of Events Calendar Pro.
So please first update The Events Calendar to version 4.4.0.1, then update Events Calendar Pro to version 4.4. Once you’ve done this, if issues persist, then indeed share the link to the password-protected post and the password for it.
Also, I would recommend going into the wp-admin edit screen for that password-protected post and viewing its content in “Text” mode instead of “Visual” mode—then copy the full, un-edited contents of the “Text” mode content of the post, and paste it into a Gist at http://gist.github.com; then share a link to that Gist here.
All of that information should help us determine the reason the shortcode isn’t rendering.
Let us know what you find!
Thanks,
George