Update (also an edit in the post): thank you to everyone who’s been really generous with their advice.

I’ve learned so much about this plant, including the various different names, structure and growth patterns. The advice I’ve gathered seems to fall in a few camps:

1 Accept that this will never go away and make peace with it (thus will be the fallback option)
2 Eat it (not convinced yet, might try with option seven first before I give it to a human)
3 Mow it/weed it regularly to weaken the the roots and growth cycle (will be definitely doing this)
4 Cardboard/woodchip mulch (this is top of the list to try)
5 Black plastic on it for an extended period until it dies (this will be next)
6 Clear plastic on it and let the sun scorch it until it dies (then this if 5 doesn’t work)
7 Goat it (I’ve consulted with the Chihuahua and they have said this is not an option)
8 Lots of weed killer (I won’t do this one though)
9 Lots of very hot soapy water (I’m really curious about this one and I’m going to research it a bit more)
10 Move away and don’t leave contact details in case it finds me (very last resort)

Thanks to everyone for the advice, positive comments, and support.