I organic garden, and would not use fabric in my compost due the chemicals, but I have been assured that newspaper is commonly used in todays organic composting!
I think I will try the newspaper cover and maybe put some straw on top instead of grass clippings, since we have so many weeds!
NOW, if I thought a mole might eat the fabric and choke to death, I would stuff the holes with my very best scraps....
but since there is no guarantee he would eat it, I just dug out the beds 3 feet down and put chicken wire down and an organic landscaping cloth, then filled the beds back in...darn moles....