First thing to check is for dodgy connections and decent drive to the alternator. Quite how it "couldn't possibly" be that, I don't know. The alternator is the source, and there's a bit of wire connecting it to the battery, where you're measuring.

Yes, 13.5v is a bit low, and it's worrying that KF's meter read even lower (perhaps they have "specially calibrated" ones).. If it picks up to 14 - 14.5V or so without load I wouldn't worry too much, but I suspect the fact that it went flat is indicative of a problem somewhere.
They have failed to diagnose the problem and picked the "fix" that would give them the greatest profit, knowing that if they happen to find a loose connection or corroded terminal while changing the alternator, they'll fix it and you'll be none the wiser.
£317 to change an alternator sounds steep to me.