Not sure about your preference, just indicating there's another way of doing this:

impl TryFrom<NotificationHeaders> for HeaderMap {
  type Error = error::Error;
  fn try_from(notif_headers: NotificationHeaders) -> Result<Self, Self::Error> {
    // ...
  }
}

A good alternative if you feel impl APNsClient becomes bloated and difficult to navigate

The push_type_str should be a &'static str so using HeaderValue::from_static() should be possible