I used to have goats, and they are browsers, not grazers They eat leaf here, and a bit of grass there, and they move on. To really graze it down close the the roots would require a grazer. Sheep are grazers. But even then they won't eat down to the roots to kill the grass unless they are starving. You would have to build a pen specificaly where the bed was to be, and keep them locked in there before they would eat it down, and trample the grass down to the dirt. That would not be an efficient or humane way to do it. Plus, I'm thinking the darn bermuda rhizomes would survive it anyway. I'm sure glad that stuff doesn't grow up here, lol, the quack grass is bad enough. Now if you had a pen of hogs, they would root out the roots and eat them I'm sure. They don't mind being in a small pen, of course you would have to deal with the smell, and they would need a lot of extra feed to suppliment the grass and roots. Then of course you could have pork chops and bacon all winter!
But think how good your flowers would grow!