If you leave from around the area of Key Largo
(where you'll also have options to wait for a weather window at an anchorage or marina), you'll be able to cross even under the prevailing SE conditions. The further you go north, the more you will have to wait for the wind
to veer S (not too typical for this area). For instance, crossing Miami (Govt Cut) to Bimini is almost directly E, and you have a N-flowing current, making your desired course SE, directly into the prevailing. If you leave from Key Largo
(Port Largo, Rodriguez Key, etc), you can basically take an easterly course (close reach) even on a SE wind and the current will carry you to Bimini.
My boat lives in Miami, and when I cross, I ALWAYS plan on first sailing down to Angelfish Ck (the N end of Key Largo) first and leaving from there. If I happen to have a more favorable wind and can leave from Miami, all the better, an extra day in the Bahamas.
And Bimini is a great place to clear in, and a very friendly, relatively inexpensive, and sleepy little place with several good marinas