My battery light is on why?

Yes, it will.

If you could post some pictures of the pulley that would be very helpful.

Roy

You are welcome.

Always glad to help.

Roy

There is a Battery Temp Sensor (BTS) in the bottom of the battery tray. The battery has to be removed in order to get to it. With the battery out of the way, the circular disk shaped sensor can be seen. You should be able to pull it up vertically with the connector and some of the wiring harness coming with it through the hole in the tray. If you are lucky, you won't have to remove the battery tray to get to the sensor wiring connector. Separate the sensor from the connector. The sensor is a thermistor, a resistor whose resistance varies fairly significantly due to temperature. Take the sensor to a location where the temperature is 75 - 80* and measure the resistance. A good sensor will have a resistance of 9,000 - 11,000 ohms.

If the resistance is not correct, replace the sensor. Because of the code and MIL, you probably should replace it anyway. An extra check before replacing it would be to check the voltage applied to the sensor. Connect a voltmeter across the terminals of the harness connector and turn the ignition key to the ON position. The voltage should be about 5 V. If that's not correct, then there is a problem in the connector or, worse still, the PCM.

The BTS is used to determine the battery temperature
and control battery charging rate.

Perhaps it is a blown fuse or faulty relay (or disconnected wires somewhere in the circuit).

I was working on that when I asked him to test the fusible link.

I assume that was with the engine off, right? If not, check it again.

Check fuse #26 in the engine compartment (15amp)

When you test for charging, you should be using a digital voltmeter across the battery. What voltage are you getting with the engine idling and no accessories on?

With your voltmeter attached to the battery, safely tap into the green wire going to the alternator and attach a jumper wire to it. Now with the engine idling and your voltmeter attached, momentarily ground that wire while watching the voltmeter and see if it starts to charge.

You either have a bad alternator or you tested something wrong.

No, that is the field wire. Grounding it full fields the alternator for max output.