that's how I killed all the tree vines in our little corner of the forest, chopped it at the base and then again three feet up. the vine in the tree will try to sprout new roots so it needs to be high enough it can't reach (if I remember right, they grew a foot or so down before dying). the roots didn't ever sprout a new vine, nor did the 3ft section do anything of note, but your mileage may vary.
I'd avoid salting if you can since that affects most everything. persistence with the other methods should work. might take a couple growing seasons to get it all.