That's a big block to spread wood chips on. And it would have to be very thick to suppress weeds, so I'd reckon you'd be looking at a few semi trailer loads.
By spraying with roundup or zero, you will kill everything, including the grass, and I guarantee the weeds will come back before the grass and will be an even bigger mess. And bare ground will be an erosion risk. Soil will be running everywhere with heavy rain.
If it was mine, I'd spray the weeds with some herbicide specifically for broadleaf weeds, as in MCPA or dicamba or starane. These chemicals probably have different 'city dude friendly' names at bunnings or such. You just need a herbicide that will kill broadleafs and not the grass. If there are bare patches, throw on some grass seed, maybe a type with runners so they will fill in the gaps, and get a decent ride on mower.
It would only cost me a few dollars in chemical to spray an acre, but I'd imagine buying the stuff from a city place, there'd be a massive mark up.
I can't see too many other options. I definately wouldn't spray with roundup, as that will just make the weeds worse when they come back, and you don't want to have to be spraying for ever more. It wouldn't take much mowing once the weeds are gone.
See ya's.