Alot probably depends on the weather the day you are leaving. I have ridden I-10 from Tucson to New Orleans. I-10 is probably your safest bet, but I have seen snow at rest area in AZ (the one with big rocks around it). I always like stopping in New Orleans. There is a Hampton Inn not far from Bourbon St. that has an enclosed secure garage.
The good news is the Tri-Glide handles snow pretty good (just like a car without ABS brakes). Keep in mind, if you do hit any snow, don't use the front brake. You will just slide. If you do use front brake and it locks up fast, counter steering will now work is the good news. Enjoy the trip. I haven't been down to bike week since the mid 90's.