I'm so excited that you guys will be making it out for your first burn! One thing to keep in mind is that there is no such thing as "dust-free." I remember thinking that I would keep certain things "dust-free" during my first burn, and mother nature showed me who was boss! It will get into EVERYTHING. Keeping your tent inside another tent doesn't mean the inside tent will necessarily remain clean, especially if there is a long dust storm.
For such a short visit, definitely use a tent. Rather than the dust-free aspect I think it would be better for you to focus on having shade/air flow and having your tent well-anchored to the ground. Air flow in your tent will make it a better place for naps and daytime visits, and you'll need somewhere shady to hang out. Even a tent-that-isn't-in-another-tent will get ovenlike and stifling during the day if it is too enclosed. I recommend that you consider bringing a tent and some sort of shade structure to go over it, whether it is some poles with camo netting or one of those party pop-ups from Target.
And keep that sucker anchored to the ground.. you don't want to see your tent flapping around 20,000 ft up in the air.
Hope this helps ya!