A living mulch is pretty common but they do often compete with the crop. I've tried clover and rye before. Clover works very well but your veggies need to be well established first for best results and your soil needs to be as weed free as possible. Clover sprouts quickly direct seeded. Not sure the brand of straw i have access to but it is gorgeous. Finely chopped, beautiful texture and a pale bright yellow. Pretty much my go-to mulch for herbs and veggies, and others like basil and greens i plant thick enough to not need a mulch but will side dress/mulch with a pro-mix during the grow season. You could initially use a pro-mix as a mulch, then try direct seeding some clover thickly on the leading edge, maybe a foot around the bed....but just trying to visualize what your garden even looks like....or what veggies you are speaking of. Some need space to produce well, so competition from other plantings between means less production of crop. Something fast growing and trailing that has one main root structure may work in some cases, like the sweet potato vines or trailing nasturtiums. Re-routing the growth back into the bed if they escape. I've used them in my brussel sprouts/broccoli/cauliflowers bed before to perk up how dull those plants look most of the early summer. I have a corn area about 20x20 that did well with clover a couple years. Now will be a pumpkin/winter squash bed this year. (covered and not used last year). I've considered a living mulch, but a thick newspaper border covered with a bale of hay works so well and needs less tending and provides a soft surface off the ground for the crop to rest and ripen nice and dry during rainy weather... Google 'living mulch'. Someone may have good ideas and successful companion planting advice more current... since i first read of it years ago, my ideas and use may be outdated....

I have some love/hate groundcovers myself. I think the biggest criteria for my love/hate ratio is the combination of how it grows (fast/slow) and where it is located. i.e. not all fast growers are hated if they are in the right place I just pulled out about 100 sq ft of pachasandra because it just got everywhere. I will miss it in the winter. (pre-snowcover). I have a huge patch of bunchberry (Cornus canadensis) that spreads very quickly, but it is in a part-shade woody tree-shrub area and it really seems to fit the woodland less-manicured feel. I also suspect it would be hard to eradicate if I wanted to. It has those thin roots that go everwhere and break easily. But I love it where it is. I have creeping veronica Waterperry in a perennial bed and it stays put nicely. Love it. I also have Georgia Blue but it is much more aggressive. I have to pull it back severely twice a year. Dislike it (but not enough to remove it). I have a gold variegated one that barely grows. I moved it this year to a new place it may do better. Lamium 'Pink Nancy' wanders and blends in nicely. It pulls out easily if it wanders too far. Love it. Ajuga, love it in bloom (have Choc Chip and Crispa Metallica). Neutral otherwise. Need to have something interesting (and strong) nearby. I'm trying out Hosta Kabitan. Thymes... love the daintiness... hate that they don't suppress weeds so well. Geranium Biokova... I consider that more as a specimen perennial than a groundcover, but I can see it is somewhat of both. Love it. Sedge, Treasure Island...grows slowly, love it because it can go anywhere safely, but unless it is cut back with other perennials, it looks messy far into spring until the new growth covers the old stuff. Cutting back a groundcover is not a fun thing. I used to when I had a little, but now its out of the question. I enjoyed and saved this old thread about groundcovers in link below. I haven't read it in a while, but it might be interesting after reading this thread.

Perhaps, you asked the question elsewhere & got a reply for how to get rid of the grass. If not, I suggest you layer over the turn after mowing short. Use cardboard, 10 thickness newspaper, or office paper 10 sheets thick overlapping and top with mulch or compost several inches up. Then plant in that. Many groundcovers are fast growers, but you will need to keep mulch and patrol for weeds until they fill out. Stepables advertises many groundcovers as lawn substitutes. YOu might want to try out a few before you make a large purchase in case one doesn't grow well for you or you prefer the growing habit of another. I don't like all the groundcovers and tended pots of them are different than run loose in the ground. There should be a substitute that you'd like and you may want to install some nonplant groundcover such as rock or patio as well. Depending on how you want to use that space... We installed a small green house & sitting patio out back as part of our lawn and then expanded with paths of wood chips and additional woodland gardens. The mowing area has significantly been reduced as well as the enjoyment of the backyard increased. I enjoy a nice patch of grass, but rarely sit out on it. I do however sit at the bistro table or the bench during nice weather. I enjoy the view of the patio daily rain or shine. I would remove more grass in a heartbeat if it is another project like we did for our backyard.

Just a warning about that fabric. Weeds will make their way through it. And then you can't pull them because they get stuck in the fabric. Poison ivy, raspberry, virginia creeper are especially tough to get at through this fabric. Your area isn't very large so you can probably stay on top of it. I think if you slash the fabric you will be defeating the purpose. You could use ten sheets of newspaper under mulch. That will kill what's there, but will biodegrade. And you can replace it whenever you put on more mulch. I have an area 12-15 feet deep by 200 feet wide, installed by a previous owner. The area was neglected and I'm slowly restoring it. Just to add to the fun, red stones cover the fabric in between blue rug juniper and myrtle, instead of mulch. And it's a slope, so at the bottom of the slope there are over 8 inches of stones that have washed down.