I agree that your success would be limited unless you had the ability to move a very large root system. The time of year is no problem since it is dormant. The problem is that the stress created by seriously damaging the root system would be complicated by winter desiccation. Also, sun stress on a plant on the NW side of a house is minimal. What it really has is wind stress in the winter. If it looks stressed I would give it winter wind protection (either antidesiccant spray or a burlap screen). Also, I would test the soil. Rhododendrons need acid soil. I would also think of fertilizing if there is a lack of vigor. Be cautious, rhododendron don't like too much fertilizer, just enough to restore vigor. They also need well drained soil. If this area is heavy clay or moist, then the plant would be struggling. If you are serious about moving this plant. I would root prune now. That means going around the plant with a spade and cutting down to contain the roots in a space you can include in a root ball when you move it. This will cause the roots to form a more dense mass that can be more easily transplanted. Then next fall or the following spring, the plant would be ready to move. You can transplant a rhododendron any time if you can preserve the integrity of the root ball and water it after moving until the shock of moving has gone. It is just easier to do when the plant is dormant. Here is a link that might be useful: Rhododendron Culture

I have seen trees grown in situ that have never developed the so called typical root system. it's not the growing or transplant system that is used that determines how the root system establishes. water tables, soil profile changes, bedrock, these are at least as important as anything else. and together, they are far more important. I've seen trees grown I assume from seed that never developed a "typical" branched root system. so do I then say that growing from seed is a prediction of failure? I guess there are many hurricanes that would agree with this. now for the part that is going to mess with folks. 3 gal is a shrub size. somewhat dependent on species, 2" is small. I do deal with a few trees as small as 7g, but these are limited to very few species/cultivars. I deal with more bareroot trees in a 2" caliper than I do 7g containers. knowing how long a plant has been in any particular size of pot is more important than what size it is. for instance, a 6' crab received bareroot and put in a 10g container in the spring is going to present few problems if purchased and planted that fall, even the next spring. or a sugar maple received 2 1/2" bareroot and put in a 40g can. get into two and half years in a pot, and yeah, there are going to be issues. for that matter, the some time frame presents the same problems, from a quart Anderson tree band on up.

Trees from seed might be a little ambitious for major results anytime soon - I get regular volunteers from my spruce trees, and some from sugar maples but have no idea how long the seed might have been in the ground before germinating and becoming a "real" transplantable tree. It's doubtful they will sprout and grow with the same speed as, say, a package of annual flowers. For your purposes, maybe whatever money and effort you want to spend would get you farther at Menard's or any of the big boxes - Menard's especially always has a big supply of Black Hills Spruce, Austrian Pines, etc. in one gallon sizes on up - last season the BHS and AP's were $5 and change for the gallon size. 5 or 6 trees would at least get you started. I'd pose this question on either/both the Conifers and Tree forums. Be prepared for them to tell you water, water, water is the key, though. (Seems like Iowa generally gets plenty of rain - so far we have too so I don't engage in any regular watering program for anything and stuff thrives assuming ground is prepared properly and the material is planted well.)

I've moved a semi mature tree peony and have nothing much to add except, dig the new big hole first, and then make it even bigger! Try to keep the soil level the same where you can, and also, it may need more water than usual during the first summer following the move. Peonies of all varieties are pretty tough, it might sulk for a year but they tend to bounce back, particularly if you're moving it to an area with conditions.