If you use an underquilt, you can put the sleeping pad between the underquilt and the hammock.
There is another trick that works: take a rectangle of light fabric (I used 0.6 oz. polyester taffeta lining material.) 4x6.5 feet, fold a few inches over at each corner, then stitched drawstring casings at the ends and edges - by simply folding the fabric over into wide hems. 3/8" elastic goes into the casings, stitched in place in the end casings so about 18 inches in the center of each end casing does not scrunch. I open the hems on the sides 2 feet toward the center and let the elastic out there and install a simple plastic slider on each of the 4 elastic lines where they exit the casings (2 feet down the sides)(bar tark on each side of the elastic opening.) You could use cord locks instead of sliders. The loops of elastic at each corner then attach to a single mitten hook at each end of the hammock to make this rig into a sub-hammock or sling under the main hammock. Leave enough elastic to adjust it to be firm but not breaking-tight when you get into the hammock. The sub-hammock holds the sleeping pad when the weather is not too cool and the underquilt when it gets colder. No more squirming around trying to get the pad under the cold spots. The sub-hammock weighs 5.5 ounces.
So, my rig consists of an adequate sleeping pad to use under the underquilt between the sub-hammock and the main hammock. Then I can get by with a lighter underquilt and have a pad when I go to ground. That's the rig I use for cold weather: hammock, sub-hammock, thin sleeping pad (also pack liner), under quilt, top quilt, cape/tarp. I usually start using the underquilt below around 45F to 50F.
Down or synthetic? I use down most of the time. Under some conditions anything will get a little wet. Conventional wisdom holds that synthetics will insulate better than down when wet. Maybe so, maybe no. If any insulation gets wet the water absorbs lots of heat and it is heavy enough to cause even synthetic insulation to lose loft. I use down for river trips, too. It seems to keep on working for me. I think down has to get really really wet before it stops lofting. Its biggest enemy is soap or detergent residue from inadequate rinsing. I just about never wet out a sleeping bag to clean it. Just wipe it down with alcohol.
You might want to experiment a bit with the 'inflatable down tube' idea before you commit to it whole hog. You will still have to encase the down in fabric inside the plastic tubes. Otherwise it will decorate the woods. Any you will have to use a pump to inflate them. Otherwise your breath will certainly wet the down. To make them self-inflating, you will have to overstuff them considerably, an expensive proposition both in cost and weight.