Last part first....No, you can't split it. Ironically, Rhododendrons internal plumbing means that each branch effectively has it's own root (own part of the rootball to itself) but the resulting damage caused by splitting open the main stem would only lead to disease and death. You can layer branches by pegging down a stem, though this still takes at least a year to root.

Moving is perfectly possible so long as the bush is manageable in size. The roots of all ericaceous plants are fine and thread-like and form a dense shallow root ball that extends out to, or nearly as wide as the crown above. If you can move that, you should be fine and the best time is basically when it's not growing and the ground is not frozen so early spring or mid-late autumn. Mulch heavily around the plant after planting with either composted bark or ericaceous compost or even leaf mould if available. Water frequently the following summer too during hot dry spells. A good soak once a week being preferable. The heavy mulch should help reduce water loss.