Blackberries have developed the secret to eternal life. Mow them and they sprout. Till or otherwise excavate and broken off roots and rhizomes sprout new plants.
The only thing that works well is chemical warfare. My local guru says Crossbow in the spring and Roundup in the fall. I haven't tried much Crossbow, but Roundup in the fall before a hard freeze does work. But since physiological process have slowed down in the fall you won't see results very quickly. I sprayed some in late August and suddenly in mid November they were dead. If they are drought stressed in the fall, spraying then won't work so well.
A friend sprayed a lot with 2,4-D in late summer-early fall and it sure worked for him, even tho it isn't supposed to work on blackberries. The canes are dead,dead,dead. Don't know about the roots. And I don't remember the formulation.
When I was doing a little forestry consulting I had a client who had 12 ft. tall noble firs that had been planted for Christmas trees and then abandoned. Blackberries were all entwined up to the upper foot or so of the trees. My logger had a trackhoe on site, so he went in and pulled up the berries, opening up the area so the 70 year old lady could go in and spray. she did that and a couple years later the berries were 99% gone and the trees were doing well.