When I backpack in the snow, I usually take a foam mattress AND my inflatable mattress to maximize the R value and provide more separation from the ice and snow.
Don't have a dog to worry about. I have had some slow leaks on the trail (last half of the JMT in '20, for instance). Unfortunately, I've never found a foam pad that works for me by itself (I'm a side sleeper).
I only recently became aware that people sometimes bring both inflatable and foam. What is the thinking? That an inflatable is just too risky by itself? Get a hole and you are basically sleeping straight on the snow? Or is there more to it than that?