Hello Chris

From MSU Diagnostician Angie Tenney: “Hand removal is the best way to get rid of it if you are not going to treat it with herbicides. You need to remove all the roots and rhizomes or they will send up new plants. You can always try to tarp it for an extended period.

Another non-chemical option: Goats would be a great option. They will only remove the top growth so I would maybe tarp it after they graze the area or bring them back if it starts growing back. https://smallfarms.cornell.edu/2011/04/using-goats-for-vegetation-management-in-the-northeast/

For a chemical option: Spot treating with glyphosate is probably the best option if you can find it.

When using products containing glyphosate there are a few important points to consider. First, as with any pesticide, remember to read and follow all labeled instructions. Second, glyphosate is a broad-spectrum herbicide, meaning it will injure or kill other plants contacted during application, so care is needed to avoid green plant material, exposed roots, and injured bark of desired plants. Third, glyphosate is relatively safe in the environment when used as labeled. It adsorbs strongly to soil in most cases (i.e., clay and organic matter), allowing even sensitive crops to be planted shortly after application; meaning no carryover issues are expected. Fourth, glyphosate alone can take up to 14 days to show full activity under ideal growing conditions. Retreatment of the area may be needed depending on the degree of infestation. Glyphosate is most effective for perennial control in the fall but can be applied anytime the plants are actively growing (temperatures consistently above 50F, when lawns need regular mowing is a good indicator). For woody species, periods of heavy sap flow in the spring may not be effective times to treat. Finally, be sure that the product you choose has only the active ingredient glyphosate or glyphosate + pelargonic acid. Products with additional active ingredients may have other unwanted effects and may delay the planting of other plants in the coming season(s).”

Here are some bulletins for you, should you need them:

https://hyg.ipm.illinois.edu/article.php?id=899

http://www.misin.msu.edu/facts/detail/?project=&id=54&cname=Bishop%27s%20goutweed

Tarping: https://weedcut.ipm.ucanr.edu/management-practices/tarping/#gsc.tab=0

This is wonderful information! Thank you thank you thank you!!!

LOL, you are very welcome
If you hunt online, there may be a goat rental in your area.(There are some here in the southern LP)
Mowing down to soil level,  and bagging, then tarping would replace the goats; since you have a relatively small area to manage. Tarping can take a couple years, and you do need to monitor the edges so nothing escapes.
Best of luck!
Laura