Clay is always too nice to me.
FWIW, the CCRM is not your issue.
When you remove the alt's reg plug, you remove the grounding pathway from the battery light. This is why the light goes out. Your alt can't possibly function this way either.
Resistance checks in alt wiring are all not efficient, nor helpful.
Do some quick voltage checks. The regulator is labeled A, S, I (the letters are embossed on the reg).
A is battery voltage. If this is missing, check the alt fuse in the underhood fusebox.
I is the signal to turn the alt on. It should show 1.0+ volt while the car idles.
S should float about 1/2 battery voltage.
The one time resistance/continuity checks are useful is to check the S circuit if you have an issue there. Disconnect the reg and the little lonely plug and check continuity between both wiring pigtails (S and the lonely little guy). You should see about 1 Ohm or less.
I would really inspect the fusible links that Clay noted. Maybe a wrench hit something while doing the balancer (not accusing anyone of anything but I often try to think of what someone could have done while doing an R/R). The links on my car were on the driver's side of the PS pump housing, just before the cable goes to the PDC (underhood fusebox).
In general, check the wiring harness as it leaves the alt. The insulation can vulcanize and the wires break, short together, etc. If you have to 'chip' the wire loom off, expect to find vulcanized wiring underneath. It's a hot climate and underhood heat thing.
I've covered this stuff in greater detail before (you might find more detail in other posts) but this is the broad strokes.
Good luck.