I have just had a quick pace out on my raised beds at home and I have about the same area of path. Mine are gravel and I got it delivered in one of those big white ton dumpy bags.
If you look at it as a volume, rather than a weight, I reckon a dumpy bag full would do it, but if you can get 2, I would. You need a couple of inches in depth and any spare can be stacked and stored. There is always a use for it. The paths will need topping up for a start.
I wouldn't bother with membrane. Annual weeds pull out easily and I use a small spray of weedkiller on anything else