I take it you're wanting to avoid "below freezing" conditions and therefore winter? (Can't blame ya!)
I'd say start in the spring. That way, you won't have to worry about winter conditions in the northerly parts of the trip; it'll only get warmer as time goes by and should be nice by the time you get there. If you wait until the fall, you may find yourself fretting about the snow forecast (and cold starts and whether you can get the engine pre-heated and that kind of thing) when you get into places like Buffalo or Montana. Hopefully not too sweltering yet in the south.
On my circumnavigation, I managed to depart Montana just as the first snow of the year was beginning to fall. Bozeman was the coldest "cold start" of the trip, just above freezing but not by much. That was the first week of October, and fortunately the next destination was California...
(My plane had a Tanis heater, but I had left the wing covers and engine blanket at home, since I had structured the trip to be "summer in the north, winter in the south".)
By the time I got to Texas, it was December, and the locals told me that the worst of the storm season (Oct) was over. That being said, while giving a talk in a pub in Dallas, we got hammered by hail and lightning (worst storm of the trip) with tornado warnings going off, and I was very grateful to be in
@Ravioli 's hangar...
I spent the spring in West Virginia, where there was still snow in the mountains in March, but everything was green and lush by May. Working my way back through Kentucky in June/July, it seems I was doing a lot of waiting out storms.
My guess is that trying to predict storms, or predict when the Rockies can be crossed, is a fun statistical exercise, but ultimately just a fun exercise. Build sufficient flexibility into the planning, and you'll be able to wait out storms and wait for a good mountain-crossing day. Nothing will work out as you expect! On my trip, for instance, I soared through the Rockies with relative ease four times, but the one attempt to cross the Appalachians had me stranded for several days waiting for the insane wind to abate.
Three weeks sounds like a tight schedule. My trip logged 84 days of flying, although *many* of these were day trips with friends for lunch, or getting through Canada both to and from... if I count just the days "putting on the miles", say, from Madison around and back to Madison again, it was more like 35-40 flying days. But also I was taking my time, and exploring and visiting friends, stopping a lot of extra places, and only flying a couple of hours most of those days. No so much going for mileage each day, as it sounds like you'll be. Beware fatigue; exploring unfamiliar places and unfamiliar weather requires more mental energy. I could last about six flying days in a row before I needed a day off or two.
Blog:
https://beetlejuiceadventure.wordpress.com
My trip was, I would say, the most fun I've had in my entire life... adventure and exploration in one of its purest forms. HAVE FUN!
