Been down that long road. I have replaced my alternator 3 times, alternator plug,alternator wires ect ect i have replaced litteraly everything that has to do with the alternator/battery.
Okay so this is where i would start. I dont believe the alternator plug would cause that big of an issue besides MAYBE the dash light indicator. I have done in shop testing with a guy who has a bench machine and if the plug is not hooked up you might get a dash light but it will still be charging. There indeed was a problem with my plug from where it actually plugged into the back of the alternator but up to today i still didnt fix it and i am not having problems. Things to look at if the car is dying would be the alternator wires, bad alternator, bad battery or bad battery terminals those are the 4 main eletrical compents that keep the car running. i would start with the big 3 since it seems like you upgraded your alternator for running a heavy sound system, the alternator positive might be corroded especially if u have a minor PS leak. Then i would check the battery terminals and possibly change them out if they look old(cheap fix 15$). in my case, it was a bad battery even if in shops it tested good. If all else fails i would say to look for a used oem alternator (got mine for 80$ on here) the most expensive thing to replace would be the battery. Change it out and return it if does not solve the problem.CHECK your battery and terminals to start, your getting a battery light so it probably has something to do with the battery either your terminals or battery itself. In my experience when my alternator went out on me i got VSC ABS light and thats when things started to shut down until i revved it up. Hope this helps