"Where's all the hikers with puppies?" Nothing about dogs in the thread title!
I use a piece of cheap closed cell foam pad, cut to 20" x 33" for my 80-lb. mostly Labrador. I fold it in thirds and it fits nicely in the pad pocket of my Six Moon Designs Comet pack (where it pads the back much better than my insulated air pad). It weighs 3.5 oz.
My dog also has a jacket (Ruffwear Cloud Chaser, recently on sale at REI) which I put on him for below-freezing nights. It weighs 7.5 oz. in a large (suitable for a Lab). While he has plenty of fur (perpetually shedding!), he's an indoor dog at home so needs some help when it's really cold. For a jacket, it's important to have it loose-fitting so it doesn't compress the dog's undercoat. Of course for dogs with thin coats and no downy undercoat, more insulation will be needed at higher temperatures.
The entire dog bed package (jacket plus pad) is 11 oz., a lot less than what you're considering.
I originally bought a dog pad made by Thermarest, the equivalent of a 36" long Z Rest. I think those have been discontinued. It is now being used by my 8-year-old granddaughter. The Gossamer Gear NightLight torso length sleeping pad would be an excellent choice for a dog, probably more comfy than the el cheapo green foam from Coleman that I use, and cheaper than buying a Z Rest and cutting it down. It weighs the same, 3.5 oz. I've seen lots of dogs carrying their own pads strapped on top of their packs, and they do fine. I carry my dog's pad only because it helps me with pack support. I give him something of mine to carry in return.
EDIT: The GG Nightlight torso pad folds flat, in thirds, so it's not particularly bulky. So does the green foam from Coleman I'm using now. Either is just the right width for any lightweight backpack that needs a pad for support (such as packs from GG and Six Moon Designs). Or, as mentioned, it can be strapped on top of your dog's pack.
Despite my clipping my dog's claws every two weeks, I keep him well away from my insulated air pad and would never put him on an inflatable pad!