What is this monster vine that is attacking my dogwood tree?
We think this looks like a type of wild grape. You should be able to gain control of this by either mechanical (cutting, digging) or chemical methods.
Your rock salt paste is not likely to work. It may "burn" a bit now but it will not kill the weed throughout the root area, and it will come back. You could continually cut the vine as it begins to grow again, or treat it with the non-selective herbicide glyphosate. It can be very carefully applied with a paintbrush to just the cut surface of the stem, immediately after the cut is made. This is a very safe way to kill the vine, without needing to use a lot of chemical or contact non-target plants and organisms.