Hi Lea, 

We are having a hard time IDing the plant. Can you provide additional photos? Maybe of the plant in the ground and the any branches or if it has flowers? That may be helpful. 

In terms of getting rid of it, what you're doing is probably all you can be and it will just take persistence. If you were to put in the work of digging up the whole garden and miss some of the roots, it sounds like it will just regrow, which is common for aggressive plants. Using Roundup, which we understand your concern close to the water, is a good way to help with the killing the roots. Just be cautious of when you spray, don't do it on a windy day to help prevent drift. You could even try painting it on the leaves and Roundup actually binds to the soil after the roots dissolve so it shouldn't be leaching into the water as long as the soil isn't eroded in a heavy rain storm. Using a systemic herbicide like Roundup or a similar glyphosate herbicide, and painting it on any leaves you see, maybe more than once if it doesn't seem to take, then going back and hand pulling any plants you see return, could be a good way to keep after the invader. 

Feel free to send additional photos of the vine and we will try to come up with a positive ID for you. 

Emily