Identifying and Transplanting Walking Iris (Neomarica gracilis)

Your description—flat, symmetrical leaves; a single flattened stem with a tightly wrapped bud that opens to a purple flower lasting ~10 days, followed by complete dieback—matches Neomarica gracilis (walking iris). It’s likely at the edge of its hardiness zone, causing winter dieback.

You can transplant it now if your soil stays warm through October; otherwise, wait until spring. It’s a fascinating plant that “walks” by rooting its flower stems.