I think you give the bindweed too much respect, you need to sow and plant as per normal, just keep pulling the new bindweed out as it appears. If you grow no vegetables, the bindweed has no competition and grows bigger.
I have a similar bed of six inches compost on bindweed and I sowed carrots in it with great harvests, but sure I needed to pull bindweed too. That was last year: this year I grew carrots again in the same bed (after a two inch top up of cow manure) and again had excellent harvests AND the bindweed this year is noticeably weaker, from regular pulling. I compost it all as well.
With respect to cardboard, it has little mulching effect when placed under compost because it is the moist all the time and soft, so roots pass through it. Main mulch effect is on the surface for clearing thick weeds, and not growing any vegetables at that time for a few months.