reconnect the battery and pull the positive wire that goes to the alternator off to know for sure man. Altho trying to troubleshoot one problem when you know you have another isnt very productive. If the alternator needs to be replaced then why not replace it and see if the issue is gone in lue with low charging. Or take it to advance and have them load test the battery and alternator.

And last, just because a battery is brand new doesnt mean that it cant be bad. Factory defects are becoming more and more common with the cheap out sourced labor these days.