My neighbors cut theirs nearly to the ground every 10 or so years and they grow back quite nicely. I have plenty of room so I just let mine go wild.

Rhododendrons have a shallow, matted network of very fine roots--hair-like, to be exact--and it is fairly easy (for a big healthy person, that is) to get the root ball out intact. Rhododendrons seem to like being transplanted, or at least, don't mind it and will reward you for moving them to a better place with a nice bloom the following year (at least, that is my experience).

I don't know if you could cut it back hard and transplant it in the same year. I'm sure someone with more experience might know.

Generally if they need to be cut back hard because they've outgrown their space - wrong plant, wrong place. It doesn't take that many years before it's oversized for the site again.

The very best way to get it out of the ground and increase your odds of saving it would be to root prune now in Fall, in Spring cut back and move - or in Spring just move and not cut back at all. Root pruning would be cutting around the rhododendron with a sharp spade to encourage a more compact rootball of a more manageable size to develop. Even then, with a shrub this size, moving it won't be a one man job so plan on plenty of help.

You can assume the roots are as wide as the widest branches - the width before you began removing stems. The rootball will be shallow, dense, with a few larger anchoring roots. I don't recommend hard pruning this time of year - it takes about a month for dormant buds below pruning cuts to begin to sprout on smaller branches, as long as 10 weeks on main larger stems. That puts you right into winter with cool temperatures, possible frost, and the shortest daylight hours of the year - not great for encouraging healthy new growth.

Here is a link that might be useful: {{gwi:391935}}

If necessary, you can remove a great deal of material. It is a general rule to not remove over 1/3 of the leaf area each year. Pruning is generally used to control unsatisfactory height or width of a plant. I don't prune very often and try to limit pruning to plants which have a shape that is unsatisfactory or dead branches. If I want to cut trusses for bouquets, I always cut the tallest flowers since this helps keep the plant within bounds.

Severe pruning is not uncommon with rhododendrons and azaleas. A healthy plant can be cut to the ground and will usually come back. Rhododendrons and azaleas have dormant buds beneath the bark which sprout to form new growth after severe pruning. However, Richard Colbert reported that such attempts at Tyler Arboretum were only successful if the plant had enough sun light. Those in heavy shade frequently died. He recommend first opening up the shade by thinning the forest canopy. Then he recommends just removing some of the top to induce new growth at the base. Then when that new growth is established, the remainder of the top can be removed.

Here is a link that might be useful: How to grow rhododendrons and azaleas.