I hadn't heard of Tordan so I looked it up. It is recommended to kill some pretty tough old brush and trees, so I bet it will work on your greenbriar. It won't kill the grass, but it will kill other broadleaf plants that it gets on, such as perennial flowers, peonies, lilies, etc. It is systemic; goes into the roots, so can kill the underground parts of the plant.
I was going to suggest using Triclopyr, which is another systemic brush killer. Maybe this Tordan is newer.
Don't give up. It takes a while to work, but you will have better results than just something that top-kills, like round-up.