If the roads are still slippery with snow, probably waiting it out is the best option - waiting until the roads are clear of snow and salt.
In the mountain passes, if chains are required, you also have to put chains on the trailer too, otherwise your chained tow vehicle will slow down but your trailer won't and you'll highly likely end in a jackknife situation if you try and slow or stop on a slippery section. Also, if the authorities are monitoring that chains are installed, they will stop you unless you also have chains on your trailer. Authorities or not, if you put chains on your tow vehicle, you must put chains on your trailer.
I have heard of reports that if the chains are installed on the braking axles of the Airstream release while driving, they will absolutely tear up the wheel wells as the wheel wells are only light weight metal, and the chains will then start to tear up everything inside of the trailer near the wheel wheels too.
I pulled a 25' Safari from Minnesota through the Rockies and to San Diego this January, and stayed ahead of a storm. I was prepared to use chains in the mountains, however found out afterwards that professional haulers pull the Airstreams through the Southern States to avoid using snow chains on the trailer for the very reason of what happens if the chains unhook.
Not sure if the States that you discuss still use salt on the road, however that cannot be very good for the Airstream's body.