Bindweed is The Pits ! You have to keep on keeping on at it, until you win. Every time you see a shoot, pull it up. Every time you find a small piece on or in the ground, destroy it or it will quickly form more plants. Try not to snap it off, try to slowly pull from the base to take out as much of the plant as possible.
I would never use any proprietary weedkillers - especially the carcinogenic Roundup which has been going to be totally banned in France for a number of years - still not quite there yet! Anything on the ground that might be picked up and chewed by an animal - especially a pup - would give me the chills.
There are some new weed killing tools on the market which
deliver a terrific blast of heat to individual weeds and which frazzles them instantly. We have always used a mix of white vinegar, salt and washing up liquid sprayed directly onto what we want to kill, e.g. on paving, drive edges etc. In the sunshine the weed or grass gets burnt by that mixture which is non-toxic, of course. You have to redo it from time to time but it's safe......and it's cheap and kind to the environment.