I have grown several types of Clemantis but never SAC. Clemantis usually doesn't root to deep so if you cut it down, you should be able to dig it out. Watch for regrowth and use a non selective herbicide like round up on th new growth. Good luck!!

I agree you'll need a herbicide to take out whatever regowth shows up next year but you can prune off all you can and dig up the rest anytime and then just wait for new growth to pop up. The sooner you get rid of it now, the less of it you'll have to deal with later. SAC is a beautiful plant and I had one for about a decade but drought got it one year and I didn't replant it there, figuring if I did, another drought would just get the next one sooner or later. The only problem I ever had with it is that blister bugs sometimes would show up and eat it right down to bare stems, but I had it in a place where it had lots of room to grow, so I didn't mind it at all. It is not invasive in my dense red clay. I think it likely would be invasive in sandy loam or even a clayey-sandy loam mix.

I got rid of a big one some years back & it wasn't all that difficult. I cut off all the stuff growing in a chain link fence down to the base, pulled it off in a huge wad and then painted the cut stems. It was gone the next year except for a few weak stems starting up that I sprayed with herbicide. That did it. It had been there quite a while, to my eye looked like a messy eyesore & I'd nearly gag on the smell of the flowers. The dense tangled growth on top made it look worse than it actually was because all in all, I got rid of it easily--- its nothing like trumpet vine, vine milkweed or anything like that.