You need to be prepared for low of 20 F along the ST in that time frame.

Snow and colder maybe Las Cruces to Phx on the ST. But it'll still be doable if you're prepared and willing to wait out any storms. .

There should be a reasonable way to cross south of Phoenix?? Maybe riding I-8 when there is not a service road. I've never heard of anyone doing that. Or maybe Tucson to Phx along side of I-10. I do think that is possible. I-10 is verboten for bicycles there, meaning there must be an alternant route. Just speculating.