I often don't carry and don't bother to inform cops when I don't.
I have had extra cops turn up when I informed them I had a HCP and was carrying (maybe just a coincidence).
I have never, as far as I know, got preferential treatment for handing my HCP over (no moreso than not doing so). This, even on one where I though they could have let it slide.
Careful cops will always be on the lookout whether you have a HCP or not. As we all know, you don't need a HCP to carry illegally and it's the work of milliseconds to draw and fire. It's largely irrelevant to their job. (Though I do want them to be warned in advance if I am carrying).