Buns can get sore feet on wire all day so you could wire the base of the run and then lay fresh turf on top which will grown down through it, the only downside is that the run won't be moveable as it is now. As well as digging out you have to think about foxes digging in, so you will need something. You could also lay patio slabs around the edge of the run if you chose not to wire the base.
We always used to wire the base of our runs when I was a kid. In our current garden we don't bother but our garden is very secure and I mean (very), in the city centre, there are no foxes for miles and miles, and the lawn is so hard and only a few inches deep - the rest is builders rubble underneath and tree roots, and the buns only go out in it during the day, mostly when i'm at home anyway.