Bindweed should I use weed killer / vinegar/ cover ?

[imgoinginsaneinthemembrane]

I have taken over an allotment plot and it is covered in bindweed. Not something I've ever had in my garden. It regrows in a week. I have dug some of the beds and covered them in plastic, but it is growing out the sides already as the wood is rotten and the beds failing apart. I sprayed a little weed killer today on these edges, but I've not used it before and it stunk. I usually take my toddler and dog there so I'm not keen on this long term. Any suggestions on what to do? Will vinegar work or effect the soil?

Those slugs need to scoff ground elder, too. I have pulled out about 5 million miles of roots thus afternoon.