Had a similar problem with my alt. Its a problem in the alts. harness more than likely.
that light is coming not because of the battery, but because the car thinks the battery "isnt charging". There should be a "voltage sense" wire coming off the alternators harness. either its not hooked up or its not working properly.
Nothing bad should happen, just the car thinks the battery isnt charging. I rode around like that for a few weeks, and nothing went wrong....finally took a look under the hood, and the wire was disconnected.
I would definatly call Dom, and tell him your issues.