Only a guess, someone may well contradict me, but I would think as they are going into their dormant period rather than coming out of it. I wouldn't want to set them back just as they were getting going, whereas late autumn and they have a nice quiet period to settle down.
The other thing I do if I am not sure is not do everything the same, so I might do one in the Autumn, see how it went, and if it was good do the other next year, if it wasn't I would try the other in Spring, or even try the other in Spring anyway, both might work.
I would go round the clump with a fork loosening it, cut it in half with the spade, and then lift the halves with the fork. The fork is kinder to the roots around the edge, and cutting it when still in the ground will hold it steady. I have done this with agapanthus. The experts said dig it up and cut it with an axe, but that made an awful mess of it when I tried it. If it is too tough to go through you can hit the spade on the foot rest with a club hammer, that cuts well and you should get it in a couple of cuts, you can chop away for ages with an axe.
Hope that is a bit helpful.