Your first step should be to ask the university's advice. The safest plan is to enter the US as F-1 up to 30 days before the I-20 start date, go to the university first and get the travel signature, and then travel to the Bahamas, returning to the university in time for classes.
While you could risk entering as F-1 both times without getting a travel signature yet, this approach runs the risk of a major hassle. For example, you might be issued an I-515A.
A more clever approach would be to try to enter as B-2 the first time and then F-1 when you reenter from the Bahamas. The main risk of this approach would merely be if the officer disagreed with or misunderstood your intention, in which case you might be admitted in the wrong status.
In any event, your first step should be to ask the university's advice. They ought to help you with this kind of situation. It would be unusual for a university to say they can't issue a travel signature because they won't be open yet. If they really did say that, please edit your question to explain what they said.