Hello Esther, it sounds like you are being diligent and committed, and can clean up the couch eventually but thick infestations take time, and it is more difficult when the couch is around roots of bushes. Weed fabric sounds good for them.
Your idea for planting straightaway into a six inch mulch of manure and compost on cardboard is good. Use the thickest cardboard you can find or even two layers, so you will need a lot, but only for this year. Also you can use either thick cardboard or mypex fabric or black poly for paths between beds and the couch must be killed there or it will simply invade your nice beds.
To prevent couch recolonising from any edges of your plot, a long term answer is to cut thin strips of six inch wood into the edges: couch roots are mostly quite near the top. Keeping grassy edges mown short also helps. Enjoy the process.