If I were you I'd just go and bung in a new voltage regulator instead of removing the alternator and then getting it tested.

There's a picture here of the voltage regulator / alternator brush assembly (as I called it then) in Assembly (3) here =>

PeachPartsWiki: Refreshing the Alternator

A voltage regulator swap should be a quick and dirty way of seeing if that is the problem (which I suspect it is with a charging voltage of 12.04V)

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1992 W201 190E 1.8 171,000 km - Daily driver
1981 W123 300D ~ 100,000 miles / 160,000 km - project car stripped to the bone
1965 Land Rover Series 2a Station Wagon CIS recovery therapy!
1961 Volvo PV544 Bare metal rat rod-ish thing

I'm here to chat about cars and to help others - I'm not here "to always be right" like an internet warrior



Don't leave that there - I'll take it to bits!