It might work, it would be an interesting experiment. If you left a strip of grass at least as wide as your mower, and kept it mowed until the vines really got rambunctious.
While I ate lunch today I was watching a program on PBS under the Victory Garden slot where the featured expert had written a book about Weed-less gardening. He set up his garden so there was wood chips between the rows and he never tilled tne growing strip, just added and covered with mulch/compost. He used a drip irrigation system so as not to provide the weeds much water between the rows.
He also pointed out something I didn't know, and that is that manure from animals that have eaten hay grown with long acting herbicides to kill broad leaf weeds in the hayfield can still contain enough of the herbicide to kill or stunt some of the garden plants. It may take a couple of years in the compost pile for that manure not to be toxic!
To find out if your manure-containing compost can cause a problem he recommended running a test on some bean seeds in pots with and without the manure before killing your whole veggie patch. He had obviously learned this the hard way. I have never asked my hay supplier what he did, my guess nothing, but his soy beans are sure weed free.