I don't know how many plants you need to attack, but poison oak grows very much the same way you have described for your companula. They way I get rid of them, is to cut back the main plant, swab the stem with alcohol because PO emits a sap as soon as it it cut, paint it with concentrated round up and cover it with a black plastic bag for the summer. Not attractive, but it does kill the plant and all of the roots it spreads out.
I only have used this method on PO because I am highly allergic. Good grief, you should see me when I go after the PO. It looks like I am wearing a hazmat suit. I don't dare let it get started in my garden. Each year, the birds do their part to start new plants.
Smiles,
Lyn
