I just completed an OKPIK weekend (snow caving) with my scouts a couple of weeks ago. As a stage setter, daytime temp was 5F, dinner at -10F, and night temp dropped to -20F. Fortunately, no wind... (BTW, an Eagle Scout Paul Siple coined the phrase "wind chill" but I digress)
The technique is to put down a tarp/vapor barrier, your camp pads and then your bag. Lesson learned is that you need to be aware of what plastic is used for a vapor barrier. We had a few boys who used plastic drop cloths that simply broke apart because of the cold.
CCF is much better, albeit bulkier, than any inflatable in cold weather. I have a Big Agnes dual air core and always find it cold below 40F. If you need to use an inflatable, I would put that on the bottom of your camp pad "sandwich".
We had a few cold toes and fingers but as long as the boys followed instructions, this was really minor under very "varsity" conditions. Hope this helps!