I'm new to vegetable growing myself, but getting blackberry down and in control i have been doing since childhood with my father (to earn a few bob as pocket money from anyone who needed it done) We started by attacking with a spade from anywhere you could get at, and it does come down quite well. Once it's down quite low, you then have to dig root out any way you can. And it might be a case of banging lengths of angle iron in the ground and levering them up, often with a brick behind the lever to stop it pushing into the soil so much. When you get going, it comes good reasonably fast, until it,s the size and shape required. You will not do the berry that's left any harm at all, in fact, it will year on year keep trying to invade the patch you are cultivating......better to get shot of the lot maybe?