I let food waste decompose in a five gallon bucket beside the back door where the DOGS can't reach it. I have a chicken-wire thing surrounding my active compost heap and it goes over the top with various pieces of wire woven through the edge to keep the DOGS from jumping in and digging even that most rotted slurry from where it is buried in the middle of the protected compost pile then covered over with a heavy layer of lawn clippings (I mulch grass back into the lawn when I mow except I catch some clippings for this purpose).

I used to use coastal hay (a type of Bermuda) in the garden but it has two problems. Seeds - you get a lot of grass sprouting in the garden from seeds that arrived with the hay. And even worse, much of the hay out there is grown with herbicides that kill broadleaf plants in the hay field. It stays with the hay or straw, it goes through the animal that eats it and is still in the droppings that you might want to compost and use in the garden. Herbicide Carryover in Hay, Manure, Compost, and Grass Clippings

Picloram, 2,4-D dicamba, clopyralid, etc. will kill your garden if it's in the hay that you use to mulch the paths, or do like Ruth Stout used to do and cover the whole garden with hay as a mulch and plant through it.