I am waging one against Japanese and Sachalin knotweeds. Actually second year, so far it looks like I might be winning but that remains to be seen,
It sounds like you have a real battle on your hands, and probably several years of work ahead. If the roots are that big and solid perhaps both a physical and chemical approach? I seem to remember that here in the uk even the bits that you dig up are handled like industrial waste, and can’t be put into household bins. Good luck!
As soon as I can get an air rifle, I will be starting a war too! The rabbits in our garden are just too much this year - we have deep holes all round the edges of our lawns, and a newly excavated burrow in our ‘wild garden’.
I have no idea of your budget (or indeed experience with air rifles), but PCP's are getting much cheaper these days. I'm an adequate shot with a springer, but a PCP off a bipod (from say, a bedroom window!) allows for almost clinical levels of accuracy - which ultimately results in humane kills. An excellent option for a 'budget' PCP is the Gamo Phox - for circa £500 you can get a kit that contains the rifle, scope, silencer, bag, and crucially, hand pump.
Thank goodness we don't have the male plant here or we'd be having to deal with it self seeding and that wqould be an even worse nightmare
That might change in the not too distant future. I read something a few weeks back that said Russian vine can cross with Japanese Knotweed and form an invasive hybrid. That, in turn, could provide a male plant that would readily pollinate the knotweed.
I'd be curious to know if the Councils have any role in sorting it out as there's an awful lot on council land around here that they do nothing about and even seem to spread it.