I would not rely heavily on the alternator to charge. That will only charge if there is excess current. Your primary charging device is the converter and should work when plugged in. There might be an issue with the converter. Did this "just happen" or are you a new owner?
PLEASE do not send questions via Private Message, my inbox fills up too quickly. Post questions in the forum as a new thread so that other users can benefit from the answers. Use Private Messages for sensitive information only (VIN, Contact info or things you do not want the public to see).