Goats are the organic way out with blackberries. They love forage, and can handle the bank. Don't leave them too long such that that "over pollute" the stream. A little us okay, a lot is problematic for those downstream. If you can, consider goat diapers or physically removing poo yourself. Additionally, goats leave the roots intact, so the bank is stabilized. It will need several "treatments" by goats. You will also need to plant some thing to take over. Not sure what is allelopathic to blackberries, but research couldn't hurt.