There are a lot of rapids, log jams, etc. on the Green River below Beaver Park. About halfway between Beaver Park and upper Green River Lake, the Highline Trail comes out of the dead lodgepole forest into open flat meadows with oxbow bends on the river, and that's where you'll want to put in. While I was camped above the upper lake nursing a sick dog a couple of years ago, a number of people were coming up the river and going back down in inflatable canoes. I was tempted to ask for a ride until I considered the combination of dog toenails and inflatable boats!
I haven't seen the river below the lakes except from a car. Beyond the National Forest Boundary you will have private property issues, but if you're prepared to cope with some rapids and maybe a little portaging, I would think you'd be OK going down to the boundary. Please note that this is a supposition on my part judging from the general terrain, not from experience on the river!
If it's not a weekend, you may have problems getting a ride. You might have to arrange with the Great Outdoors Shop for their shuttle to pick you up. Since there's no cell phone reception until you're almost to Pinedale, you would have to arrange this in advance.
There are lots and lots of lakes up in the high country waiting for your packraft! Enjoy!