Important: Apply the foam in a well ventilated area!
I ended up using "Great Stuff" spray foam intended to fill small (1-2") gaps. I tried the foam for larger gaps, but it didn't seem to make much of a difference. In fact, I thought it was harder to work with. Each can cost about $2.50 and for the size of this mountain, I went through 7 cans total.
Hold the can close to the wire frame and begin applying it in rows. The foam will expand slightly after application so you don't have to pile it up. You will need more cans of spray foam then you originally estimated. It looks like it goes a long way, but unfortunately does not. Keep in mind that once you open the can it will need to be used completely.
The outside of the foam will harden in a couple of hours and become completely solid in about 24 hours. Don't rush it --- you don't want to cut into the gooey center.