Yes, both "화요일에 뵈요" and "뵙겠습니다" are fine, 뵙겠습니다 being more formal. However, 뵈요 should be 뵈어요 or 봬요 (뵈다 -> 뵈어요 -> 봬요) grammatically. 뵈다 and 봬다 are pronounced very similar, and I think native speakers say 뵈요 more often than 봬요. But the correct form is 뵈어요 or 봬요 (뵙겠습니다 is always 뵙겠습니다 since -겠습니다 form doesn't have the 아/어 syllable in it). 봐요 is the plain form of 뵈요 without the sense of respect. It will sound rude unless the professor is your personal friend or much younger than you. So, I would use "네, 교수님, 화요일에 봬요 [뵙겠습니다]".

교수님 화요일에 봐요.