After running a cable directly from the front power distribution center to the starter bypassing the fusible link, it shut off while driving.

Hello, this system is not controlled by resistance, it is controlled by a pulse width modulated signal on the Gen Field Control circuit from the PCM. Take out the external regulator, which is only adding resistance to a wire that you are probably shorting to ground. The PCM also varies the amount of charging control based on the Ambient temperature sensor, there is a fusible link on the red wire that goes to the Starter and off of that stud the wire runs to the Battery, Ill post a diagram of the circuit. The PCMs Gen field control is around 250Hz pulse width control, but it will change depending on electrical load. But you should also check that the ambient temperature sensor is working correctly too, if its outputting incorrect signal the PCM will change the charging system output based on that temperature. Ill post a circuit of that too,
This Ambient temperature sensor is wired into a control module that is on the one of the data bus networks in the vehicle, you should be able to check its output with a scan tool on the PCM live data PIDs, To check it, if you unplug it there should be 5 volts on the Signal wire while unplugged. The other wire is a Ground, but to check the data bus you would need to use an oscilloscope to see the network transmitting data, so it's just easier to check the PCM live data and look for the Ambient Temperature sensor, it will most likely read a voltage and an outside temperature.
Now, you added an external voltage regulator, so I don't know how that is wired up, if you ran a resistor between the Gen Field Control wire and to Ground, it most likely overheated the Alternator and possibly the diode trio inside. So, when you get things hooked up correctly, you should check for any AC voltage ripple coming out of the Alternator. It should be less than 50mv at idle, some allow for a little bit higher when there is more electrical load, but at this point it's difficult to say what has burned out by now.
You should do a voltage drop test from the red wire post on the Alternator to Battery positive at idle and see how much voltage is being lost on that wire. Do the same for the negative side of the circuit, from the Alternator case to Battery negative.
To see the 250Hz control signal from the Alternator you would also need a scope for that, but there should also be live data from the PCM for that on a scan tool.

It wouldn't hurt to verify the engine grounds to the battery negative as well, seeing that the engine was out. Check it the same way at idle, from the engine block to the battery neg, and the body to battery negative. There should be very low voltage reading, below a 1 volt. You might be missing the chassis or engine grounds. This would cause the PCM to lose it grounds as well.

https://www.2carpros.com/articles/how-to-check-a-car-alternator

https://www.2carpros.com/articles/how-to-use-a-voltmeter