Anything outcompete pampas lily-of-the-valley?

Salpichroa origanifolia (pampas lily-of-the-valley) has a stronghold across my backyard as well as neighbours yards. It’s considered a highly invasive species here. While we’ve ripped up a chunk of it and regularly mow the above ground portion, there are so many underground rhizomes/runners and seed-bank creating new plants…

I’ve been in the property for less than 6 months so don’t know when it was established, however seeing how overrun the neighbours yards are on both sides with it I figure it’s been here a fair while and crept across properties

I’d love to take a “work with nature” approach and plant something that could outcompete it, or create conditions to halt it in its track.

Suburban block in Melbourne, australia, dry sandy soil, growing in both sunny and shady spots, in a patchy weedy lawn.

Any ideas? 🙏

Edit: rest assured I am working on swapping the suburban lawn for a more diverse and permaculture informed garden 😀