as it's growing "on" a tree, you've got a root system supporting the vines that are climbing the tree.
here's one way to deal with it.....
using protective measures, clip / trim / cut off the leaves and cut the vine back to a manageable size but _not_ to the ground.
you may have saved, or you can get, from your local florist, those cute little plastic vials with a plastic cap - used for keeping cut stem roses / flowers "in water"
fill the vial with RoundUp, stick the end of a poison ivy vine through the cap and into the vial - let it suck up all the RoundUp it wants . . .