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RE: I just looked up the internal impedance

It seems to me that the battery does not care about its internal impedance; it cares about the impedance across its external pos and neg terminals. Since an intact capacitor should have near infinite impedance at DC (although large value electrolytics do leak a bit, on the order of micro-amps), there is essentially no load or very little load. Uncle Stu wrote about this, at least tangentially; if the capacitor or the battery get hot or if your battery wears out prematurely, that means you've most likely got a leaky capacitor. Maybe I am still misunderstanding the issue. Sorry, if that's the case.

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