Probably not. In my experience, Blackberry bushes will continue to grow until they crawl out from under the covering to root wherever the cane touches the ground. A better way is to mow it down repeatedly to exaust the plant and to prevent seed production. Spraying with an herbicide is more effective method of control. Glyphosate products (Roundup, etc.) should be applied in September and October when the canes still have foliage and berries are present. Just be sure to do it before the first hard frost of the season. Triclopyr products (Crossbow, etc.) are also effective. Be sure to follow the herbicide instructions carefully.
It might be possible for you to "rent" a goat somewhere. There are people who do this, and goats LOVE blackberries. However, I doubt if they pull out the roots, so you'd be back to where you started once the plants started growing again.
I'll refer you to two WSU approved sites for more information.