The problem with any long-distance route is that it is bound to include both low-traffic roads and high-traffic roads. It's simply hard to get from A to B when A and B are far apart without going on some busy roads. One solution to this is to find a base area and do day rides or small loop routes from there. Then you can pick only the quiet roads in that area. In a month, however, you might run out of good places to ride in one area. So you could pick several bases, and use a vehicle to get from base to base.
I do have a question. Do you want low-traffic roads primarily for safety or for peace and quiet? I assume both, but compromises always need to be made. There are some roads that are fairly low-traffic roads that are a bit dangerous, and some high-traffic road that are pretty safe. One reason is that most low-traffic roads have little to no shoulder, which means that you are at risk from the small amount of traffic that there may be. And many high-traffic roads have very wide shoulders (e.g., interstate highways).
If it's peace and quiet that you're after, pick some place with low populations. That description applies to most of the West except for Southern California. But you also said that you were interested in climbing. In many areas of the West that have a lot of climbing (i.e., mountainous), it's also very difficult to build roads so they don't build too many of them.
There are many rural areas of the West in farming and ranching country that have low traffic and lots of roads, but they don't offer too much climbing. I'm thinking of places in Kansas, eastern Colorado, Nebraska, the Dakotas and Oklahoma.
One other idea is places like Appalachia and the Ozarks. There is plenty of climbing there, and not too many people, so not too much traffic.
