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They're lovely posts aren't they. I have one. And I've got a set of both types of clamps. I use oval carbon rails on mine so the spares I have are the same as the ones already on your post.
But the point of my comment is to confirm that both types are definitely available. I either called (or emailed) Hope directly as I couldn't find them on line either and they couldn't have been more helpful. Or just ask your LBS to order them for you.
Yes oval rails are oval to have added strength in the vertical dimension whilst maintaining clamping diameter of a regular steel rail when viewed from top or bottom.
Side loaders won’t work generally so my guess would be no, one or the other needs to go.