Okay, we are NOT talking about the Hemi tick some hear from the engine. Matter of fact, my Redeye has no Hemi tick. I have owned a '06 6.4L with loud Hemi tick, my '15 was negligible on tick, several of our eight Jeep's including my current 5.7L has Hemi tick, the Redeye has none. I posted a quick video in my channel a few months ago, because this one don't tick.
The OP clearly describes what we are talking about. MINE DONT GO AWAY AFTER ITS WARM. I listened last night after a pleasing drive with the car. It is probably as benign as a Hemi tick, but it is a noise that seems like it comes from the right side exhaust under the passenger seat maybe just behind it. It is not possible to hear it except at idle. But OP, mine is identical to your description.