The limited luck I've had with taprooted weedy perennial's is to keep any plant material cut off at the soil line and dig down to remove as much of the taproot as is possible. That will be quite a chore with the area you have.
If I've cleared a small area of weeds
with taproots, I place a large plastic trash can cover over the spot to block out light for a few days at a time. The cover is lightweight enough that the soil doesn't compact and is easier to lift up to access soil condition then mulch covered landscape fabric. If you see any above ground growth, repeat the process and cover again for a few days.
You could invite an herbalist over to dig up their year's supply of burdock root.