You have the main mesh with a hole of a specific shape. All of the hole's vertices lie on the same plane.
You have a separate mesh (smaller) where one of its protrusions has an open hole, also of the same shape and size as the main mesh's hole, but is NOT co-planar with the main mesh's hole, and its plane isn't even parallel with the main mesh's hole's plane.
I can select both identical loops on both meshes that I wish to merge somehow, but I want the main mesh to stay put, but the smaller mesh to actually move intact into position. Bridging doesn't work because that adds geometry. Merge first/last/center/cursor won't work because those would alter either or both meshes. Merging would move only selected edges, which is not what I want.