If you had a double layer hammock you could stick it in there and no sliding problem...
but if not.......
If you can manage it i would be impressed
I've had really good luck with a wool blanket from Pendelton on the inside of my single layer hammock. It doesn't compress at all, so it would be hard to carry backpacking, but for nights around the house or car camping, I prefer that to CCF.
tie off two corners of the blanket to the ends of your hammock and fold the other two in.
I have a wool shirt with a Gore Windstopper lining. It is crazy warm. Warm, to the point that it is unwearable on all but the coldest of days.
A layer of Tyvek sewn to the outside of a wool blanket might prove surprising.
generally the advantage to wool is that it breaths so well which allows it too be used in a large temperature range adding tyvek or the windstopper membrain kinda defeats that....
Has anyone considered quilt batting? There are several options other than the fluffy white stuff that first comes to mind (cotton, wool, silk, bamboo). It can be easily cut to desired size, and is generally quite lightweight, yet very warm, and inexpensive. Of course it could be sandwiched between some nylon, but if it's inside the hammock that may not be necessary, wouldn't be waterproof if you didn't though.
As for the 'slippage' - I have no idea without actually trying it. If you work with a cotton batting (think super thin) you could sew little patches of some sort of non-slip item to one side of the batting in 3 or 4 key areas - like maybe the shelf-liner non-slip stuff (again, super light weight).
I'm leaning toward a Warbonnet blackbird - at 5'1" I'm wondering if I could get one made slightly shorter, don't need the length and extra weight. Haven't found any, but are there 'petite' size/weight hammocks?