Dead end on "Update Apple ID Settings" Prompt

I've been getting the "Some account services require you to sign in again" prompt for days now, with no way of actually following the prompt - I click through on the main settings page, but there's no "continue" button, and tapping the prompt itself does nothing (see screenshot). I tried turning the phone off and on again, and manually signing out and back in, and the badge and prompt are both still there. It's starting to really irritate me - has anyone else been experiencing this?