I'll be the voice of caution here.
Will the new plants have
enough time to get established before the hard frosts of winter come? My fear is that those cuttings will freeze all the way through and the ongoing freezing and thawing over he next 5 months will kill those cuttings before they get a chance to root.
If it were me, I'd leave the plant in the ground for the winter, and then go ahead and spit it come spring time. Or, if you've got a sun room or south-facing porch where it doesn't get so cold, perhaps plant those root chunks in well-draining potting mix (like a 50/50 mix of cactus mix and potting soil) and let them overwinter in a warmer place. You aren't going to get much more than a couple of weeks of growing season before freezing weather renders them a dormant, so dividing them and planting them now just feels like there is no advantage whatsoever.
Just because it might possibly survive the winter doesn't seem like a good enough reason to do it now. Patience, grasshopper. It'll only take you 20 minutes to do the very same thing next spring.
"The rule of no realm is mine. But all worthy things that are in peril as the world now stands, these are my care. And for my part, I shall not wholly fail in my task if anything that passes through this night can still grow fairer or bear fruit and flower again in days to come. For I too am a steward. Did you not know?" Gandolf