How to kill a Guinnea Vine without poison?

I have an old Climbing Guinnea Vine with very long roots that has grown underneath the house's paving and is interfering with the pipes. So it has to go.

Spraying it with chemical poison from the hardware store is the easiest option. But is there an environmentally-friendly alternative? It's not worth spending $6,000 to re-line the pipes when a $15 poison can stop the problem.

I can't dig it out as it's inaccessible. Cutting it to the ground won't work as it has many access points for suckers to re-grow along difficult areas near the fence. I'm a bit lost as to what to do.

sometimes you have to do what you have to do

If roots are already in the pipes, just killing the plant isn't going to make those roots already in the pipes go away.
That is why there are companies like Roto-Rooter.

Pulling the plant and pulling it every time it tries to make a comeback will use up the energy reserves and the roots will die from energy expenditure with no new energy coming in.
This takes a while but it will work without any poisons needing to be used.

I've dug as deep as I could and ripped out the vine deeper than a foot below the soil surface. If I see any new vines I'll rip them out when young and what's underground will turn into compost. Thanks.

The professional gardener told my parents it's impossible to get rid of without poison and even then it might not work. We'll see if this natural way works better.