I collect Himalayan blackberry fruits to eat every year, but I’ve never considered eating any other part of the plant. In all my reading, I’ve never encountered any reference to someone eating the root (or the root tuber). Therefore I have to say no, the roots aren’t edible, simply because I don’t whether they are safe or dangerous.
Many plants have one part that is edible and another part that is unsafe to eat. The edibility and deliciousness of Himalayan blackberry fruits doesn’t mean that the roots are safe. Plants or parts of plants shouldn’t be eaten unless there is definite evidence that they are safe.