I have been there a couple of times recently, most recently last summer for the exact reason that you are going. Since you are going to shoot the Milky Way I assume that you are there at or near the new moon, as I was, and the astronomers tend to go there at that time as well and a lot of the campsites were occupied with those folks, but I was able to get a spot. And if the campground proper does happen to be full, the jeep roads that head west from the camground had plenty of (I assume legal) spots for dispersed camping with no amenities other than an amazing view.
You prob know that there are lots of Utah Juniper at the campground but you have to drive a ways to get to the Bristlecones. Anywhere from maybe 20 minutes to over an hour, depending on your grove of choice. I had checked in at the temporary visitor's center, and noticed signs all over the place indicating "No Overnight Camping" within the boundaries of the Ancient Bristlecone Pine Forest. So I ask the Ranger...um...I am going to be out there all night, shooting the trees and the milky way...whats the difference between "camping" and just hanging out there all night? To which he replied..."Well, I am not going to be there checking on you!"
So I would do it again any time, and I think they see a lot of us types out there at night... just be sensible about being out there overnight...strive for less than minimal impact..I did have dinner there, slept, and had fun when my alarm went off at 2 AM...wish I had some help as it seems you will...and as da bird would say...no fire rings!