I speak from personal experience. I have gone directly between the Chesapeake and Marsh Harbour four times on my own boat, and a dozen or so times on delivery usually in colder months. Lots of deliveries have been owner-aboard from folks that are tired of the ICW, don't want to cope with the anchoring and tax issues in Florida, or just want to get to the Bahamas.
Weather forecasts out four days are reasonably reliable and out two days are pretty good. That's plenty of time. It only takes four or five days from Little Creek to North Man of War Cut. You should still have a way to update weather forecasts underway. Weather fax using an inexpensive SSB, a laptop, and a patch cord works fine.
Note that there is nothing about "rounding Hatteras" in this route. That isn't to say that Hatteras weather systems aren't of concern - they can be big. Those systems don't appear from nowhere though. Stage in Little Creek with high speed Internet and wait for a window. The windows are bigger than conventional descriptions would indicate. I've only been hung up twice.
From Chesapeake Light I head 135T (Diamond Shoals at Hatteras would be 165+/-) until I'm across the Gulf Stream - watch the water temperature to peak and then drop - when the temperature settles out (usually about 10 or 15 degrees F above the coastal side) your through and can head for Green Turtle or Marsh Harbour. Note that you get as much fair Easting as foul Northing from the Gulf Stream at this latitude. A day and a half and you are across. Then it's just another three days to the Bahamas, warmer all the way.
I love watching boats head down the ICW and run into them many weeks later while they are entering the Bahamas when I have been there for a month or more. *grin* Warped sense of humor I guess.
Warm clothes and heat are important. There are three effective options I can think of: 1. a diesel heater, 2. a Mr. Buddy or similar propane heater, or 3. a car heater core spliced into the engine cooling circuit (which means running the engine for heat).
Good technical undergarments are the difference between comfort and misery. One full set plus two extra pairs of socks will do. Some intermediate layer you already have and then offshore foulies will get you through watches. Outer layers keep you dry. Inner layers keep you warm. Below should be comfortably warm or you just won't be having any fun.
Plan good hot meals for your crew. A cold sandwich on a cold day is nothing to look forward to, no matter how much hot coffee you pump out.