Hello...I have included a response below from the University of Maine to the same question you have since it is a very good description of what you can do to control this weed without plastic or chemicals.

And, as for a suggestion of what to plant instead, you might want to try some Golden Alexander. You may have to purchase it on-line if you can't find it locally but it would be great plant to have. Also, I have included the link to our website that lists many other native plants that could be options for you. Good luck with garden!  Robin

https://cornellbotanicgardens.org/plant/golden-alexanders/

https://extension.umd.edu/resource/recommended-native-plants-maryland/

University of Maine information on Bishops weed/Goutweed: 

https://extension.umaine.edu/gardening/2022/05/10/removing-bishops-weed/

Repeated pulling of bishop’s weed can be effective, but it can take a lot of time and persistence depending on the size patch you’re dealing with. It has a very persistent root system that will sprout new growth from very small segments. The key is to remain committed to pulling/cutting it very regularly (weekly) and not let it get a foothold at any point in the process of eradication. As soon as it’s allowed to photosynthesize for a period of time, then it will get reestablished. It’s a war, not a battle.

Start with hand-pulling efforts for a few weeks, using a garden fork to loosen the area around the plants to get as much of the roots as possible without breakage. It’s especially helpful to do this when the soil is moist as the roots will more easily pull through the soil without breaking. If the area is mowable, mow the area for a few weeks after pulling at the lowest setting as your next step. After that, cover the area with a layer of cardboard covered with mulch to fully suppress the remaining growth. If you’re working around perennials, your best bet is to lift the desired plants and carefully remove any goutweed roots that are growing through the perennial root mass. Pot up any perennials or temporarily plant them in a holding bed until you fully eradicate the goutweed from the area (1 or 2 seasons). Keep an eye out for at least 2-3 years for stray goutweed plants. They can appear for many years in severely infested areas.

Be aware that bishop’s weed/goutweed may cause skin irritation to some individuals. It’s best to wear gloves if you have sensitive skin.