Battery indicator light

Yes, it could be the alternator or a wiring problem. Here is a how to for checking it. Do this and let me know what you find:

https://www.2carpros.com/how_to/how_to_test_alternator.htm

If the unit is bad this guide will walk you through the repair.

https://www.2carpros.com/articles/how-to-replace-an-alternator

Please run down these guides and report back.

How was the charging system tested, and what were the results? This problem is acting up intermittently. Was the testing done while the problem was occurring?

There's a lot more involved with this type of problem. Most simple testers only check that the battery voltage is between 13.75 and 14.75 volts with the engine running. You can do that yourself with an inexpensive digital voltmeter. If that voltage is okay, that only means it is okay to perform the rest of the tests, but you need a professional load tester for that. Specifically, we need to know "full-load output current" and "ripple voltage".

All AC generators have at lest six diodes, and if one fails, all you'll be able to get is exactly one third of the rated output current. 30 amps from the common 90-amp generator is not enough to run the entire electrical system under all conditions. The voltage regulator will detect that missing one third of the output and turn on the "Battery" warning light in response.

If it is necessary to replace the generator, also look at the age of the battery. Due to the design, these generators develop a lot of harmful voltage spikes that can damage those diodes, and the internal voltage regulator, and they can interfere with computer sensor signals. The battery is the key component in damping and absorbing those spikes, but as they age and the lead flakes off the plates, they lose their ability to do that. It is common to go through four to six replacement generators in the life of the vehicle. To reduce the high number of repeat failures, always replace the battery at the same time, unless it is less than about two years old. If it tests good, it can still be used in an '86 or older GM car with the older generator design.

Sounds like your alternator is not always charging have you had your alternator tested yet?

OK when it acts up are you watching the charging voltage at the battery?I have seen bad ecms cause random charging issues. Had a 2004 Chevy cavalier do that and it was a bad ecm because it controls the alternator charging.

I can't speak to the difficulty, but starting in '87 with this design, GM generators have an extremely high failure rate. They develop huge voltage spikes that can destroy the internal diodes, voltage regulator, and interfere with computer sensor signals. To reduce the number of repeat failures, replace the battery at the same time unless it is less than about two years old. As they age and the lead flakes off the plates, they lose their ability to dampen and absorb those spikes.

You're right about the charging voltage. When it's working properly, you'll find between 13.75 and 14.75 volts with the engine running.

GM never designed a vehicle with an eye toward service. Their concern is how it can go together the fastest on the assembly line. Also, there is no rhyme or reason as to where they mount things. The generator can be in one location in one vehicle, and a totally different location on the same size engine in a different vehicle.

I looked up the instructions for yours and found there's a lot of variables that I and probably you can't answer. You're going to have to look for yourself to see if you want to attempt this job. For the choices I selected, there are 16 steps in the list of instructions that include removing engine mounts to roll the engine forward to access some bolts, remove the drive belt tensioner, remove the generator's bracket, remove the power steering pump with a warning to not disconnect the hoses, and other steps.

There's also a tip posted that suggests lowering the engine cradle with the engine on it to make generator removal easier. If you want to try that, don't start until I explain what can happen to handling and steering response, and how to avoid those miserable problems.

I wasn't able to copy and paste the instructions. Your best bet is to get a copy of the manufacturer's service manual, then read through the steps with the line drawings to see if you want to attempt this job.