Unless the snow situation changes and we get a wet spring, you aren't going to have a ton of snow to worry about, even in early June. Thus the GPS may not be necessary. You could always use your cell phone's GPS in a pinch. I'd rather carry the Tom Harrison JMT Map Pack and a small compass to navigate as you won't need to carry the batteries. The JMT trail is pretty easy to follow for the most part providing you have a map to know which way to go at junctions that aren't Marked with JMT (as some just point to a destination that you'll find on your maps). The tops of the passes may be under snow, but as long as you know where the pass is in snow,you'll usually just head straight up to it once the trail becomes burried. Even in early June, you'll share the trail with numerous PCT Thru-hikers going north so there will likely be foot prints to follow and it may not be necessary to carry the SPOT either since you'll see other hikers.
Depending on how much snow is on the ground when you start, you may need an ice axe. If the snow remains really low, then probably not.
My favorite sun shirt is one by railriders. They make a couple that have a mesh strip down the torso side's and along the shirt arms. Eco Mesh-TOp, Adventure Top, Madison River Shirt, Versa-Tac Light shirt. Some come pretreated with permithrin (Insect Shield) for the Mosquitos. As I mentioned earlier, I like my Tiley's LTM-6 Airflo hat for the sun, but it does cost. The brim stays pretty stiff in rain and it has a double wind strap (one behind the head) which helps keep it on your head in wind better then just a chin strap. Otherwise, just get something with a wide brim that isn't too floppy.
Unless you tend to be cold all the time, I don't think you need both the fleece and down jacket. Just one or the other will do.
For DEET, I recommend 3M Ultrathon DEET lotion which is easy to apply in the amount you want where you want. REI sells it. A 2oz bottle weights about 2.5oz. Make sure you bring some chapstick.
Have you tried carrying the Bearvault with the CDT? I tried the Garcia Can in a ULA conduit which is an older version of the CDT and while it did just fit, I wasn't happy with how it carried. But the bearcan shape is a bit different then the bearvault. So I was just wondering how well it works.