You can try to root greenish stems but blackberry cuttings are not highly successful. Sometimes a very elaborate setup is required including covers and misting.

Dividing in summer would have a very low chance of success.

I would try digging out the smallest clump that has sent up one or more protocanes (new canes for this year), cut out all fruiting canes from this clump before digging, and get as much soil as possible (1-foot radius and 1-foot deep if possible, and transplant the whole thing into damp ground. This would have a good chance of success if the ground is kept damp well into autumn. If the protocanes are already longish, cut them back to 3 or 4 feet.

If your new garden is not available for some weeks or months, the dug clump can be kept alive in a pot. Otherwise, just buy new plants next year.

Ah, darn, I was hoping it would be simpler. I think I might take that last option and buy new plants next year. The second garden is ready, but I just have so much to do to wind up this one and also take care of my other gardens. If I can get to an easy to dig piece I will at least try a small clump.

The dug clump has a good chance of a small crop next year; with new plants it will of course take longer. If you pre-wet the intended small dig area, it will go a lot easier (sprinkler or open hose end for at least 1-2 hours).

Here is a you tube video on the subject that may help https://www.youtube.com/watch?v=h8VhLJTLvBo