Since you’re hosted by WPEngine, I recommend contacting them directly and asking for advice. It might be that your CPT isn’t reliably making it into their cache for some reason, but they can give you a better answer than any of us.
You should definitely escalate the ticket you’ve already submitted for the issue. If it’s an intermittent problem that seems to resolve when you reset the cache, then it’s directly related to the cache. And I know WPEngine uses some non-standard code … so there’s nothing we can do to help here.