Bishop weed as living mulch?

We have an area of bishop's weed along our property line where we have fruit trees and small no till beds. I've been thinking that letting the plants run their course as a living mulch would be a far wiser plan of action than weeding them out as I had been doing. The beds would then be ideal for vertical crops like pole beans, amaranth, etc. I would love to hear if anyone else has tried this.

We are experimenting with living mulches. We are looking at a mix of white and red clover on one bed, weed grass (which isn't a grass) on another and pineapple weed on another and possibly a vetch or two (I have to identify them first as some can swamp the crops). All of which are relatively low growing and can be cropped and dropped if they too much for the veggies.

As for Bishop's Weed I don't know, is this it? Wiki - Aegopodium podagraria If so I personally wouldn't use it in the veggie garden because of the way it spreads. Just my thoughts, I could be wrong, I have been before.