I suspect it is not because of the bearings but rather from the increased strain put on the alternator from the lack of voltage from teh battery...
if the battery was that dead then it wouldn't start, plus would be the same thing as when running with high beams, spot lights, stereo, etc on - would be no different (as in having the alternator under load, it would still have to provide the load current). The noise is made by the belt on the pulley (if that's what we're talking about), so it would make that noise under the same load regardless. Can also be an old serpentine belt btw. If it's bearing noise then it doesn't matter if the battery is dead or not, it would be noisier under more load but replacing the battery won't fix the bearing issue - might as well be both.