The best time to lift a Fatsia would be from March to April, before it start growing again OR at the end of the summer, September-October, when the soil is still warm and growth is slowing down.
Take cuttings in September, too late now, about 5 inches and put in coldframe.
Dig a hole in advance, plant in, soil and compost and a lot of water. Reduce some of the leaves (reduce water loss)