Battery light

They can be removed by pulling them straight out with needle nose pliers and inspected visually for a break in the filament.

Who said the charging system is okay? The battery voltage with the engine running must be between 13.75 and 14.75 volts. Yours is well below that and will not fully recharge the battery after starting the engine. You might get enough current to keep the engine running, but the "battery" light is on because system voltage is too low.

The fuse for the generator's output circuit is a large one that's bolted into the under-hood fuse box. We do not have time to waste testing fuses when you are paying us by the hour. Instead, with the engine off, measure the voltage on the the fat output wire bolted to the back of the generator. A test light will work okay for this too. You should find full battery voltage there all the time. In this case you will find it is okay. I know that because a fully-charged battery will measure 12.6 volts with the engine off. The only way you can have over 13 volts with the engine running is there is some current coming from the generator and getting to the battery, and it has to go through the fuse to get there.

A good suspect for low output current is one defective diode of the six. That will reduce the maximum output current to exactly one third of the generator's rated capacity. 30 amps from the common 90 amp generator is not enough to meet the demands of the entire electrical system under all conditions. The battery will have to make up the difference until it slowly runs down over days or weeks. You need to have the charging system tested by your mechanic, not a parts salesman. The two main tests in question are "full-load output current" and "ripple voltage". If your generator does indeed have a bad diode, ripple voltage will be very high, and output current will be very low. You need a professional load tester to measure those things. Simply measuring charging voltage is only the preliminary test. If it is okay, that just means it is okay to do the rest of the tests. Charging voltage is often still within the acceptable range when there is a bad diode in the generator.

13.84 volts just barely sneaks into the acceptable range. Again, that only means it is okay to do the important part of the tests, and that requires a professional load tester to measure "full-load output current" and "ripple" voltage. Without that equipment, you might be able to get a rough idea whether to expect to find a defective diode. With the engine running at idle, and while monitoring battery / system voltage, turn on as many high-current systems as possible. Those include the heater fan, head lights, and rear window defogger. With a bad diode, the generator will not be able to supply the needed current, so its output voltage will drop considerably. Dropping a little is normal, but the voltage should still stay above 13.75 volts. You're already close to that even before drawing that high current.