Request persistent storage permissions

If any of the above are true, the permission is already granted, and if not it's automatically rejected.

This is incorrect; the permission must still be requested. Until it is requested/granted the origin will still be treated as "best effort". The site must explicitly request the permission. You can observe this by calling persisted() prior to calling persist() on a site that has never requested the permission.

all I need is a sure-fire way to get the permissions popout to show in Chrome so the user can grant or deny that permission

Chrome does not have a permission popup for this permission, so there is no such way.

The preferred model is to not offer guarantees about offline capability until the permission is granted; that is, request the permission and only show any sort of "your data is absolutely definitely available offline" assurance once it has been granted.

As mentioned by Joshua Chrome does not have a permission popup for this permission.

What I did was manually enabled the push notification in the site settings for my website.

This was enough for Chrome to grant my website persistence storage permission.