I would try weed killer. You can get one that you dab on the leaves, and it will only kill what you dab it on. If you use a spray, it will go on the other plants as well, so I would try the 'dab on' one.
Sorry I don't know what it's called, but you can find it in any garden centre.

Other than that you simply get on and pull it out, regularly.

Or use a regular liquid systematic weedkiller and use a paint brush to apply it to the leaves.

Have to admit my first reaction was Oh dear, move house....
Tackle it from the base. In spring keep a look out for the new shoots which come fresh every year and try spraying them while they are still only a few inches high. You will have to keep on doing it time and time again because the roots go down very deep and they take a lot of persuading to give up. But by spraying it at ground level you can avoid inadvertently killing the brambles as well. Get a suit of thornproof clothing for Christmas...