What this sounds like is you have a "bad" new
alternator. I'm guessing you bought the reman one cause it was a few bucks cheaper, keep in mind that the reman one went bad once before. I think if you swap out the
alternator with a new one your problem will go away.
The only other thing is if you have an under drive pulley on the
alternator(if you robbed the
alternator off your friends
hot rod) , this would cause the same issue because it would require a higher rpm to "excite" the
alternator and begin charging.