Yes thanks I know how a battery works.
What I'm saying is that there's no full proof method for determining this kind of issue. You certainly can't do it easily with a multimeter as the voltage drop only occurs for a fraction of a second under load, and we already know it's dropping voltage somewhere.
The battery is always the most likely candidate as it's performance will wane almost as soon as you start using it, so this would not be money down the drain even if the problem lies elsewhere. For £70 it's the easiest and cheapest swap out, even just to eliminate it as the cause. The alternator should be good for at least 120k miles, and if faulty, would be displaying other symptoms. At £230+ fitted it is not a cheap or easy fix.
If the battery is suspect then the amp will be pulling short sharp bursts of current which the battery cannot supply, thus the dimming lights. A powercap in this instance will provide a reservoir to overcome this, but will ultimately be masking the real cause.