Quack grass is the bane of my universe. Pull it and half your garden bed is uprooted. It lives very happily under mulch (I have found it thriving under black plastic after months) and can cover half a foot a day with its runners. Chop it down and it will go to seed at three inches tall. Put the "dead" grass in your compost and you won't be able to turn the pile, its roots system will be so dense. Pulled runners will re-root inside a week if they're not disposed of properly.
Having said all that, I have grown huge and very healthy crops with the problem of quack grass. I pull what I can, keep it from going to seed at the edges if possible, and then just sort of put up with it. I'm sinking a border 6 inches deep around my veggie bed, but I know it won't solve the problem entirely. Can't be a perfectionist in a world with quack grass.
And it's a mercy you're not! I am, but I've learned humility.:) I won't use the poisons either, but I hear quack grass doesn't respond so well to those either. I'd chop it down with scissors/shears as best you can and hope the root weakens.
I am trying to "live in harmony" with the quack grass and other pests. I love the bees and we do use hormone traps for the wasps. I add ladybugs each year to eat the aphids on my Japanese snowball bush and sadly had to use diatomaceous earth on carpenter ants this year. Even some grass in my flower beds are welcome as long as they don't take over but the quack grass... I even put bishops weed in a bed to see if it can overrun the quack grass, but so far they are about even!
I haven't found a remedy either, but I do my best to try to make life as miserable as possible for it while not damaging too many plants that I am trying to grow. Gardening seems to be a balancing act of getting some plants to live and others to die. Keeping up with chopping the edges of my "lawn" (ha ha, it is just grass and weeds that were here when we moved in and when I watered them they turned green so I kept them) an attempt to keep the root runners from going into the beds, then pulling the grass that sprouts in the beds. It's hard stuff to discourage, but then so am I. A battle of wills.