Hi, Fiat. I have many camellias, but I've never grown one in a container, so I can't give you any advice on that, but I have moved camellias several times. When my house was being remodeled a couple of times, I had to move several that were in the way of the planned additions. I also moved camellias when some old trees were taken down and all of the shade plants suddenly had too much sun. This happened at various times of the year, and I never noticed any difference in the condition of the transplanted camellias, whether they were moved in spring, summer, fall, or winter.
I have a few camellias growing in sunny spots in my garden, but all of the transplanted ones were moved into varying degrees of shade, from dappled shade to full shade. I think lots of shade and lots of water are essential to newly planted or transplanted camellias in our California climate.