Can you identify this invader?
Actually, this is a fern. It's Sensitive fern, an unusual-looking native fern that is beneficial to birds and wildlife. https://www.marylandbiodiversity.com/view/413
It does spread, but it is a very slow spreader. Your landscape looks pretty natural (or, at least, we don't see any turf.) You could consider enjoying it as is--just another layer above the periwinkle (vinca) in a layered landscape.
It has a tough rhizome system, as you've noted, and there is really no way to selectively remove it other than digging it out. Just keep it mowed if it gets into the lawn and that should limit spread.
The periwinkle, on the other hand, is a workhorse groundcover but is not native and doesn't benefit the environment (in fact, it is listed as a non-native invasive plant because it can take over parks and natural areas.) Maybe adding some other native ferns to the patch (avoid painted fern!), and you could make this an intentional-looking fern bed.
We think we see some stiltgrass in the back of the middle photo. This is a terrible foreign invasive grass. You don't seem to have too much. There is still time to get rid of this obnoxious weed, because it is an annual weed and it has not produced seeds yet this year. If possible, pull it out soon. It has a very shallow root system. If you don't remove it, it will make seeds in late August-Early September and you'll be battling it in that bed for years.