Just prune both the blackberry and ivy back hard. Blackberries are, like raspberries, canes and you can cut them back to the ground - just remember to wear solid gloves and cover your forearms.

Fruited blackberry canes should be cut out after fruiting anyway. If you cut the lot down then if you keep cutting before it has a chance to grow again much it will eventually die. Or you could use weedkiller on the new growth. Hard to kill ivy with it though as it runs off the shiny leaves.