For instance, say the snow date is September 10th on average. You would probably want to be out of the Washington mountains by September 1st in case of snow (or be prepared to hunker down in a good tent or motel until it warms back up). Then say it takes 150 days to get from Tampa to Washington. That would mean you would need to start approximately April 1st. However, you then need to balance that out with arriving too early in Canon City, CO (case of the Colorado Rockies). Not before June 1st for sure, probably more like June 15th. You will definitely have snow at the passes but they should not be blocked. It WILL be cold at night.
One thing to consider also is to reverse your route and/or starting from different locations, i.e. start from San Diego, May 15th and go clockwise. You will just need to play with your speed, route, and weather (can you tolerate 30 degree nights?? to get something you can live with.