Getting rid of Green Alkanet (Pentaglottis sempervirens).. help

One of my customers' garden has this stuff growing all over the place in borders.

The tap roots on it are like nothing I've ever seen. They are well over an inch thick and go down longer than a trench shovel can reach. It's incredibly hard to get them out without snapping the bottom off as they reach under concrete and slabs. The lady has a dog and will not allow use of any weedkiller in her garden.

She had a roll of fibrous membrane and we doubled it up, put it over one border and weighed it down with quite large rocks, but I feel it will just lift it up or punch through it like dandelions can.

The plant struggles to grow in acidic soil so do you think a few bags of ericaceous compost will kill it off (nothing else grows in the border because this stuff has smothered everything else. If it works, Rhododendrons or Camelias could fill in the gaps. What do you reckon? Has anybody else banished this plant from a garden without chemical warfare?