The size of the alternator shouldn't be an Issue to the Battery light.
The voltage that the Alternator is producing should be that it is charging the Battery when running, the difference in it being off and it goes to 12 states that you
are probably have a 12V which is normal.
I suspect that your Exciter wire or that the Circuit isn't back biasing the bulb to turn it off.
That could either be the Bulb, the Diode.
The diode has a Positive and a Negative leg, the negative or "N" junction is denoted by, the Band on one end of the diode, for the circuit, and the flat side of the LED case and a shorter leg. Diodes can regulate if the circuit is designed as such, but in this case the Power from the Cluster on Run prior to start is such that Current flows
in the circuit as the Ground comes up from the pin2 (brown) wire on the cluster through the black diode causing the lamp or led to turn on.
When the Car is started and the Alternator detects the positive voltage on the Lamp it builds a field and starts charging..The charge from the alternator being greater voltage than the 12V in from the Black wire pin 13 causes the Same voltage to be applied to the bulb and since the bulb or LED has the same voltage on both sides it turns off.
Since yours doesn't then you need to concentrate on the Diode (not the led). It could also be that your new alternator does it have a PLUG or 2 Studs and uses ring connectors? as if you had to remove the plug and convert then you may have an issue with the "blue" wire on the alternator they do go bad...
Key on in run not start there should be a voltage on that wire (blue) if you disconnect the Plug, and measure the blue wire to ground.
This diagram is for a Caddy, but I think it is wrong.
Here is a Westy Rabbit.
There isn't a Diode in the Circuit but the LED in either, so while the Diode is working then suspect that the BLUE wire has a Break in it and is floating.
Some voltage regulators use internal 12v feed to start the field to start charging via the 12V line into the regulator from the batter directly which could be your case as the alternator starts charging it sends a Sense Voltage to the LED to turn it off. I suspect that your blue sense (exciter) wire has a break in it at either the Plug on the Alternator (quite common) or at the cluster side.
One way to test is to jumper 12V to the Blue wire on the cluster to see if the bulb will turn off as I suspect it will.
Since you have upgraded to 120A Alternator you may want to upgrade the Alternator plug to Ceramic for the increased current that you are producing as the old plug which is Rated for 55 amps may not live long.
You can use insulated Female Spad connectors and do away with the plug by using separate wires to the 3 pins, I would use red for the Positive ones and a ble one for the exciter wire...
https://www.serial-kombi.com/en-GB/...cal-/-fuse-box-relays-and-wiring-looms-n38553
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http://innovativewiring.com/?page_id=3314