I would definitely get the perennial weeds out if you can, or use the area with fewer.
You could spray with Roundup - I know its not organic, and I don't use it now on cropping areas, only did at the start, but is one of the weedkillers that doesn't persist in the soil. It's so hard to weed asparagus beds after the event - you really do need to prepare well beforehand to avoid disturbing them once in.
When I did my asparagus bed, because I'm on heavy clay, I dug in loads of sand - play sand, not builders sand - and it has made such a difference in that area. I get good crops, and the others on our site struggle where they haven't.