Yeah, I would agree with Lazur. Try converting that one arc shape (made with Ellipse tool) to a path. Select and do Path menu > Object to path (or whatever the key combination is).
Technically, it is not a path, it is a shape (in vector language). So SC would not be able to cut on that, because it's not a path. I don't know, maybe SC tries to convert it to a path, and somehow that causes it to be shifted. Or maybe SC just doesn't know what to do with it, at all, since it's not a path.
Also, you may want to do a couple of node edits, to connect the whole outline into a single path (First ungroup everything, then highlight overlapping nodes, then "Join selected nodes" button on Node tool control bar) (or whatever the key shortcut is). I'm guessing it will cut better if it has 1 single path to cut on. I could give you more detailed steps, if you need them. But it sounds like you're pretty much getting it already
However, if that doesn't fix the problem. Try the 2nd layer drawing. There is a well known bug that causes objects to shift their position in certain situations; and drawing on a new layer usually solves it.
Let us know how it goes