Raspberry/rhubarb bed overrun w weeds

When we moved into our house 2 and a half years ago, we were excited by the raspberries and rhubarb in an in-ground bed, raised maybe 8 inches from the rest of a yard and separated by a 2-high landscape brick wall. We wattle-fenced it off from the grass so the dogs couldn’t go in there. Problem is, we haven’t stayed on top of the weeds and now we’re overrun with creaking buttercup, herb robert and others. Some of the rhubarb is huge and we’d like to keep it, but transplanting may be an option (except fear of bring the weeds with us). We’re pretty frustrated with the whole thing and are ready to sacrifice the raspberries if that’s what it takes. They are ever-bearing and we cut canes to the ground each year anyway. They grow back bit matter what we do. We’re considering cutting canes to the ground and sheet mulching the whole area. Worried about the canes pushing up the cardboard. Advice? Other things we should consider? (Washington State, USA)