The best way is to starve them out. Cut them down before they flower. Keep taking the top growth off the plant and the plant is unable to make its own food via photosynthesis.
Another method is to bury them if there is room. No plant can withstand no air, no water, no sun.
Depending on the conditions, the environment, the commercial services spray lightly with glyphosate during the late spring, a month before flowering and while the plant is vigorously growing. Wait an entire month and then cut the entire patch to the ground. Cover with clear plastic, not black...and staple the edges securely to the ground. I would also bury the edges with shovel fulls of soil.
Note: Rake up all debris and bag carefully! Every little tiny bit has to be removed because this plant can grow from just a wee little bitty tiny piece! I'd get a shop vac! Using a blade weed wacker instead of a line trimmer would also be a good idea. The line trimmer would mangle and chop too much. The blade would cut the canes down more cleanly. Use a big heavy tarp and carefully gather the canes as you work laying them down on the tarp so you don't walk all over them and break them up. I've been doing some research on this since I answered your question the first time and WOW. I think I am understanding much more clearly your frustration! I've not had the problems I could have had, I was lucky my clients had time to wait.
Hopefully there is plenty of summer left to 'sterilize' the soil and remnants of this plant.
Towards the end of fall, remove that plastic (please!!) and dump 1 to 2 to 3 feet of topsoil over all for the winter. There is lots of other stuff to dump on top for the winter but hey, no way can a plant survive this...but THEY do!
Follow up with hula hoeing or spreading a bit of glyphosate on any new growth in the spring.
I know this 'weed' well. I am from the Pacific Northwest. I've been a licensed pesticide applicator for decades and there is no herbicide that is able to get this under control within one year like this more manual management. Think of it as job security. Educate your clients about this very noxious exotic weed. Break your services down into manageable packages.
This knot weed is not your enemy. It is just a smart plant. Your Cooperative Extension Service in...Massachuchetts (sp) grins...is your best friend! Get to know them. Do you have access to human poo/sawdust decomposed mulch? This would be your favorite rich uncle.
What you've been doing is very close to the best you can do. Spray only when the plant is in vigorous, early season growth (glyphosate). Do not spray so much you get drip and certainly not during any stiff breeze. Allow the glyphosate to kill the roots; at least 3 weeks.
Then cut the plants down to nothing. Make sure you get every little piece of debris picked up.
Solarization is done with clear plastic, not black.
Remove the plastic during the fall and bury the best you are able. I used the Power Company's chips to get rid of blackberries, almost as bad as Knot Weed. 6' piles, 10' wide. Added alfalfa pellets for nitrogen. Steamed all winter long. Kind of cool.
In the spring those 6' piles were 6 ", seriously, and I never saw another blackberry in that area.
Don't forget weeds are just plants. Weeds somehow have gotten this reputation and becoming everyone's top monster in the garden. I don't get it. Weeds just are not at all a problem. Regular maintenance, a little at a time, starving them out, smothering weeds and/or using a mulch without any weed seeds...
Note: I've just learned that the flowers don't produce viable seed, so you can stop worrying about the timing, this plant is simply amazing at vegetative reproduction, however. Don't mess with patches of this plant along waterways. You have to alert the Weed Cops (Cooperative Extension Service) and they are licensed for pesticides and water. The only way to eradicate (?) this plant is a multi weapons approach and glyphosate is important.
Glyphosate is safe to use if you follow the directions. More is not better. Perhaps multiple applications but not spraying every inch. There is one mother root system generally. Whatever part of the plant you spray is translocated to that mother. Killing the roots by interrupting some major systems. Quite brilliant. It is not a horrible toxic chemical. It will not stay in the soil. You most certainly can grow vegetables in that soil some day in the future.
This plant was brought to the states in 1850? Something like that and it is now in every state and in Canada and in Mexico and is a huge problem. This is one plant that warrants using a Pesticide. Hands down. Forget vinegar. Forget hot water. I wouldn't go so far as injecting glyphosate into the canes. I am still researching that bit. Makes no sense because the translocation process is how glyphosate works.
I just had an idea about the debris. A ghost buster type flame torch! After allowing glyphosate to do its job, I wonder if torching SMALL patches would be effective? I think you'd LOVE that method, huh? Grins!
There is also some Patent Pending! method? Chemical? to eradicate Japanese Knotweed! That should make you happy as well. It is called 4 Stage Model. I've got to check out what that is all about!
the Best way to control Japanese...