From the look of it, I think it is more likely a Viburnum tinus than a bay. You can cut V tinus off at the ground-level, and it will regrow. You can dig them out to stop them regrowing, but they have very hard wood (including roots) so aren't terribly easy. (They won't resprout from root fragments like bindweed and ground elder.)

You can also clip them back very hard to size, if you want to. When I moved in, there was an overgrown 8 foot hedge of them, and I cut it back almost to ground level, dug some of it out, and am now maintaining the regrowth as a 3ft hedge. After you have cut it back hard, you won't get any flowers next season. In general, pruning needs to be done immediately after flowering if you are to get flowering the following season. The flowers are worth it, long lived and aromatic, and can come in winter.

If it is a bay (you should be able to tell from the smell of crushing a leaf, though v tinus also has aromatic leaves, it is more curry-like though), then we successfully moved a small bay recently, because the builder couldn't build the new garage with it in the way. In fact we moved it twice: it was put into a temporary location for about 9 months before being moved to its final location. For us, it was easy: the builder just dug it out with the digger he was using for the foundations. The rootball was of a size I could easily lift it and carry it. It has re-established fine. It was moved after being 7 years in the ground, but, that said, is rather smaller than your bush.

And bays can also be clipped like a hedge to down-size them, though I would be uncertain whether they would come back from such drastic cutting as can be done on a viburnum.