Is it possible to just cut it at the ground, even if it's rooted in multiple places? If you can manage that, in spring when it starts to come to life, cut it 1" from the ground and immediately paint the cut stem with the highest concentration of glyphosate you can find. Do not dilute it at all, you need it full strength. If you wait even 5 minutes, the stem will start to callous over and won't absorb the glyphosate. It's absorbed down into the roots this way to kill it. I found a 43% concentrate on Amazon that I still have, it lasts forever done this way.
I've been doing this to get rid of trumpet vine that was growing between daylilies, peonies, hostas and other flowers and it doesn't harm anything else in the soil near it, and these plants are right up next to it, no space between them. I know most try to stay away from harmful chemicals, me included, but doing it this is very targeted and only kills that stem down to the roots where you applied it.
