I've always found it easy to weed ivy, no thorns and shallow roots. Or you could put down a groundcover sheet, and the piggies will still have the space to explote, and you can do grass deliveries several times a day.
Ivy should be fairly easy to rip up. We put ours out for 30 minutes while we cleaned the Pighouse. There was a buttercup patch that I tried to remove (got most) and trusted my adult pigs to not eat it. They are pretty sensible though I'd never trust them 100%
I'm a bit odd in that I find ivy quite satisfying to pull up. It usually consists of quite long stems which rip up really easily, so you can actually clear a lot in a short time. I once cleared an area of about half a tennis court in a couple of hours. The worst thing is thinking about it, once you start it's not that bad.