아니면 and 이나/나 means "or" as you said. and your example sentences are pretty good! But in that case, you'd better not to use particles. Or, if you want to use a particle "-로" is proper.
Another, when you speak to a teacher, Korean shows more respect. Korean add the higher level of respect form "-시". But the best word in the honorific form "to drink" is not 마시다 but 드시다.
1. 선생님, 차나 커피 하실래요?
2. 선생님, 차 아니면 커피로 하실래요?
the best option: 선생님, 차 아니면 커피(차나 커피) 어떤 걸로 드실래요?