The most direct route between Boston and Colorado is not a southern route  It goes through Chicago and is 1800 miles (27 hours or about 7 hours a day for 4 days).  There is no way to do both the most direct route and the southern route.  Trying to go south of that route will add miles and hours and may or may not avoid snow.  The best thing to do IMO is to watch the weather and pick a snow-free weather window to do the trip.  And try to figure out a way to schedule the trip so that you can be flexible with the time of departure, and the number of days.
A slightly longer route (36 hours) would be Boston down to Harrisburg and then down I-81 to spend the night in the area of Charlottesville (this part of the route is the most likely to have snow IMHO).  Then to Nashville for the second night, and Dallas for the third night.  That route will have you driving about 9 hours a day.  You will not have any time or energy for sightseeing - dog friendly or not.
As for dog friendly hotels - you can do that on Google maps as easily as I can.