I promise you, there is no way landscape fabric is going to solve this problem. Fortunately I've never had the misfortune to battle this weed, one of the most notoriously difficult ones to eradicate. But I think you are on the right track with smothering since you've removed the desirable plants.
There are weeds, and then there are
weeds!. A
weed like this is an all-or-nothing affair. Leaving any tiny bit is unacceptable since it will never stop taking over until you win the battle. Gardening around it isn't an option. However, once you gain control of your garden, unless you hardly ever go look at it, you should be able to keep 'normal' weeds under control by simply pulling anything whenever you are taking a minute to look at your pretty garden, hopefully often. That way, there's never 'weeding to do,' a horrible situation to let happen, and much more time consuming overall.
Smothering blocks the light, and physically prevents plants from being able to push up out of the ground. When a plant is as determined as chameleon, you have to bring out the big guns. It's going to take something strong, that is left alone long enough that you know it can't possibly still be alive -
definitely not newspaper.
I would use sheet metal for this, if at all possible. Nothing can grow through sheet metal, but it has the drawback of not being able to secure the seams well, sheets are only a couple feet wide. You can't cover it with straw because the straw will just blow away. It could be covered with mulch though, or raked leaves.
A less strong option would be *thick* plastic, like a tarp. Plastic has the drawback of being ugly, but you would need to put at least 6" of mulch (heavy, shredded hardwood - not something light-weight, like bark chips or pine straw) over it anyway for it to have any hope of working. Plastic has the advantage of being large, easy to work with, fewer seams for plants to possibly exploit.
Cardboard would be my 3rd choice, but even large, flattened boxes from furniture or appliances also have seams that plants can exploit if not overlapping well - like at least a foot, and a single layer may decompose before the plant dies under it. I've never failed to kill anything I smothered with cardboard, even English ivy, berry vines, but would use 2, maybe 3 layers of it, from reading so many anecdotes about this plant' tenacity, and knowing it may take longer than the time required for 1 layer to decompose. Cardboard would also need a thick layer of heavy mulch to complete the light-blocking aspect, keep cardboard from blowing away, and make it heavy enough to physically prevent the plants from being able to break up through it. However, if done well, overlapping the seams so nothing can escape and keep going, the cardboard doesn't need to be removed later. Just add desirable plants again after you've determined, for sure, the chameleon plant is dead.
If you think you have a problem now, this is what landscape fabric looks like after a few years. About 4-5 years after being laid, there are tons of weeds, it looks hideous, and although I pleaded with Mom not to do this, *I'm* the one who will have to tear it out - and all of the weeds and baby trees. You can't spray failed landscape fabric with anything to get rid of it.
P.S. For future reference regarding less hard-to-kill wayward plants... Herbicides that go on foliage won't do anything if used after pulling, that's kind of backwards. The leaves absorb the chemical and it is delivered to the roots. Spraying anything but healthy leaves is ineffective. I don't think glyphosate (Roundup) kills chameleon plant at all, if it even makes it ill - or it wouldn't be such a notorious weed. Folks would just spray it and move on, not writing articles, blogs and journals about how it won't die! For any plant that it *is* possible to kill with RU, it works best when it's a warm, sunny day, sprayed on happily growing plant material. Except for the possible, unusually warm day, it's way past RU-spraying season (if one expects it to work.)
Whatever you decide will be best for you - best of luck!