Battery light turns on only when braking

A common cause of this is one failed diode of the six inside the generator. You need a professional load tester to check for this. With a bad diode, you'll be able to get only a maximum of exactly one third of the generator's rated current. You didn't bother to list the engine size so I can't look up the size of your generator, but for the common 90 amp unit, 30 amps is not enough to run the entire electrical system under all conditions. Turning on additional loads, such as the brake lights, will result in the electrical system voltage dropping. That will turn on the "Battery" warning light. Also, since all generators lose efficiency at lower speeds, the system voltage will drop when idling too.

The load test will show maximum output current and "ripple" voltage. Those will show if there's a defective diode in the generator.

Thank you very much for the quick reply. Sorry I didn't post the size of my engine because I don't know it, but your answer is very helpful. Does this likely mean the 6 month old alternator is likely to go completely soon?

Diodes don't fail real often. When you get a replacement generator, a new one would be horribly expensive, so you always get a rebuilt unit. Since those diodes don't fail often, and since they're relatively difficult / time-consuming to replace, they don't get replaced. That means even though your generator is only six months old, it has parts in it that are original. Normally just the brushes and bearings are replaced, and everything else is reused.

Diodes can fail anytime but it's more likely to occur when they get older. That's what your symptoms point to, but don't take my word for it. The load test will tell whether I'm right or wrong.