If you have something like FS or farm spraying company by where you live, you could pull some up and take it in there and show it to them. They can tell you what you need to kill it and if you specify that you dont want to touch the lawn grass they may have something or can mix it up for you on the spot. It might be a little more expensive then jugs of roundup but if they know their stuff they can steer you in the right direction.
There is an app I use on the phone called plantsnap that will let you take a picture of the plant leaf and will identify the plant for you (may take several attempts to get a correct match, and the app has annoying ads trying to sell you to upgrade) Then type in the plant name online and what to kill it with and you should get some results on what herbicide to use.
2-4-D is a broadleaf herbicide that shouldnt affect most lawn grass
My viney plant is called winter creeper, thats the stuff that no other herbicide would touch until I bought the crossbow herbicide. That stuff finally killed it