Green Alkanet taking over

I’m pretty new to gardening so I’d appreciate advice.

At the back of my garden a patch of Green Alkanet has popped up and is taking over.

I managed to take the flowers off before they seeded this year and tried to dig some out. The plants seem to be well established and have come back up, and I know the soil has been compacted and unused for years.

I have some garden beds to remove (pretty sure some of the roots are under them), but then the plan of action is to:

Dig out all of the soil and plants to a depth of 20cm (nothing else grows in this area).

Put cardboard down.

Put some soil on top

Put something down, most likely a clover/grass mix. (This is more to cover the ground, so it doesn’t lie fallow and to provide something for bees this year- it isn’t the long term plan for this part of the garden).

Is there anything else I should be doing? Any idea on how to get rid of the soil I’ll be digging out? Our council website doesn’t clarify if this counts as green waste or not (I’d assume not).