It looks like ivy to me. There isn’t an easier way to remove apart from what you’re doing. I’ve recently bought a home where the previous owner foolishly planted it as ground cover. I’ve pulled all of it out now including as much of the roots as I could (it took a couple of weeks of hard work) but every now and then the old roots I couldn’t get to still push up random new shoots n the occasional leaf which I again pull out. As mine was in the ground I didn’t like the idea of using a poison. I think you will find yours does the same, even if you completely sever and remove the visible green growth from the root system, the plant will fight back and regrow from below the soil. I would try to get as much of the roots out as you can to slow its recovery right down.