Hi natyananda! Came out of lurking to reply to your post, haha. I'm waiting on the Fulbright for Bangladesh and am an alternate for CLS Bangla as well so I'm doing some nail-biting over your question myself. I don't think we have any definite orientation dates for either the Fulbright or the CLS. But looking at past dates, the Fulbright orientation falls during the time one would be abroad for CLS. So the two may conflict and if they do, you might have to decide between the two.
The CLS website states:
For applicants to the Fulbright U.S. Student Program Applicants to the CLS Program may apply for and receive both a CLS Program award and a Fulbright U.S. Student Program award in the same academic year. However, participants may not accept both a CLS Program award and the Fulbright Critical Language Enhancement Award, which provides supplemental critical language study funding for Fulbright U.S. Student Program grantees, in the same academic year.
If a participant receives a Fulbright grant with a mandatory pre-departure orientation in Washington, D.C. that falls during the CLS Program period, he or she may need to choose between accepting the CLS Program award and the Fulbright grant.
Hope this helps! And best of luck to you