There are dozens of places to cross the Rockies - it depends on your preference.
Here's a short synopsis of route and climate -
Technically, the Rockies begin in Santa Fe, NM and stretch all the way up almost to the Yukon in Canada. But if you are going from Missouri to Los Angeles, I think that Banff would be out of your way - even Glacier National Park, which is simply stunning. It sounds like it would be best to head through southern Colorado or northern New Mexico if you are heading to L.A. Be forewarned, the prevailing winds are southwesterly in the Four Corners regions, so you are likely to be bucking headwinds much of the trip.
In addition there is a series of landforms you will cross like books on a bookshelf that alternate cold and hot in the summertime. The Rockies have late springs and early falls and cool summers. The Great Basin has moderate springs and falls and hot summers. The Mojave Desert has summer temps in spring and fall and blistering temps in summer. If you cross the High Sierra, you have a climate regime similar to the Rockies. The Central Valley is extremely hot in summer. The coast is cool.
That said, I would advise crossing the Rockies no earlier than mid-June. Why cross the Rockies if you are trying to get through them as quickly as possible and cannot enjoy them? Late June is wildflower season and is truly spectacular. But then you would be looking at a late-June / early-July crossing of the Great Basin. Since that is the case, it is better to be north than south of the Grand Canyon. If you were to follow the Western express to Cedar City, Utah - you could continue across to Tonopah, Nevada - then head south to Los Angeles via Bishop and US 395. That route avoids the worst extremes of the Mojave Desert. Another option. is to continue from Tonopah to Yosemite National Park - truly worth the ride - and cross over to the coast and down. That would be further, but would have very little riding in the hottest areas.
You might think that a late trip could sneak you through the Rockies in early fall and get you into the hotter regions when they have cooled off. But the days are really getting short after the fall equinox and California is at the end of its summer dry period - thus many park and forest areas are closed due to fire danger or actual fires.
Weatherwise, you can get snow well into June and early snows in September in the Rockies. You really need to be extremely cautious in May or October. In addition, snow depth is asymetric - early snows in September melt quickly, but snow remains at higher elevations into June and July.
My experience is that there is an abrupt weather shift sometime in mid June from a high rainfall pattern to occasional afternoon thunderstorms. The Rockies fall into the Great Plains precipitation pattern - with half of all annual precipitation occurring in May and June. Similarly, in the early fall you may get a snow or two in late August or early September - but they are usually only dustings. But after about the 20th something changes. Some years it is later, some years a little earlier. But after the 20th of September, the window for touring closes rapidly.
Have you considered it the other way?
Because leaving from California in May and heading east is almost ideal. Perfect in Calif.; lovely in the desert - especially if it has been a cool, wet winter; and you get to the Rockies just in time for the wildflowers.
Colorado - West Elk Mountains
Nevada - Railroad Valley & Kawich Range
Now, don't get me wrong.
I live in Wyoming and I ride all year.
But I live in Wyoming.
Best - J