Not circular at all. Use one set of data (let's assume your Sargasso sea mud would be relevant) to plot a line. Date the points along that line using ANOTHER dating method. Then, take a measurement of the C14:C12 ratio in your non-mud sample. See where it falls on the line you just made.
Yeah, that is really over simplified. Not all that for off from what actually happens, though.
And yes, radiocarbon dating is only good for a relatively small swath of time. Useful to tell how old a piece of human history might be. Pretty useless for anything pre-historic. Depending on what you are trying to date, there might be better techniques. If you REALLY want to pin something down, you would use several techniques on several samples.
I see better what you're saying now. Still, the problem is the assumption of known C12/14 ratio. We believe, but cannot prove, the ratio a thousand years ago was the same as it is now, so the difference in ratio today is refective of age (half life) rather than a difference that was already present when the object ceased fixing carbon.
To my knowledge, the Sea Mud and Ice core data sets are only used for CO2 levels and do not speak to ratio.