Your mystery underhood switch is most likely used as the sensor/trip for the
alarm system.

Man! You are working through all the correct steps to diagnose your problem,
with no joy.

'With the engine running place the VOM leads on the appropriate Battery terminals,
What voltage reading are you getting? (You're looking for a range
between 13.0 volts and 14.1 volts)[this is telling you what voltage the
Alternator is producing].
Now try the same test with the Headlights,Rear Window Defroster,
Air conditioning(With the Blower Fan on HIGH) all turned on
with the engine idling.The Voltage will drop ,but should maintain somewhere
around 12.1 to 12.5 Volts[If the Alternator cannot produce this minimum
the battery slowly drains down]

You may also have a Phantom Drain.The Multimeter in line test is valid
(Even using the Voltage scale...I've seen Amateur Wrench Turners,Who
are even cheaper than I,use a 12 Volt test light in line between the
Negative battery post and the disconnected Negative Cable to chase
down Phantoms,and THEY GOT LUCKY because there was only ONE
gross Phantom,Also because [Kinda like a Divining Rod] They claimed
the ability to judge the amount of current drain based on the Brightness
Factor of the Test Light.)
However,TOPGUN is correct, the definitive and most informative testing
is Amperage.It tells you the amount of power being drained(Milli-Amps or
whole Amps),[Small Current drain or Large].
The Voltage (or test light) in line only tells you that there is A drain.
(As others have mentioned,it might only be the Clock...OR it could be
the Clock and a Phantom...There's no way to tell)

Set your DMM/Multimeter to the DC Amperage scale(It will probably only
allow a 10 Amp Fused scale as the largest)and place it in line between
the Negative battery Post and the Negative Battery Cable.
[If you're spooky about your Phantom being larger than 10 amps and
damaging your DMM's fuse (Not Bloody Likely!) pick up a cheap DMM
at Sears (usually less than $10.00 USD) ]
Now run your "Pull the Fuse and watch the DMM scale" tests.If you're
showing less than 1 amp you can step down to the Milli-Amp scale.
The Clock,Etc will be pulling a low Milli-Amp level...Anything else is
your Phantom.

The very first thing I'd do,as soon as the DMM is hooked up in line,
is to pull the Trunk Light Bulb out of it's socket.Something about
that Circuit frequently hangs up in the closed position(usually,
it's the switch).

'Have you done Continuity testing on the Power Circuit cables?
(I.E. The Cable from Alternator Positive to the Battery Positive...OR
the Negative Battery cable [basically a Ground] )

__________________
'84 300SD sold
124.128