JFS,
I have driven the past 2 times from Lower Westchester County in NY and if you leave at 3 or even 4 am the beltway should not be an issue. I've done it 2 different ways leaving at 4 AM and also leaving at night (although many might argue that 3 or 4am is still night). By night I mean at about 9:30. Here's my reasoning...not sure where you are staying, but if you leave at 3 AM from south Jersey I'm estimating travel time of about 18-19 hours...which gets you the FLA at 10-11 AM?? Check-in at most hotels is usually not until 4PM and after a 19 hour drive even I wouldn't want to go into the parks (and I usually open and close the parks when we are there) -something to keep in mind anyway.
When we did leave at 4AM we stopped off for the night in Hilton Head, worked okay, but you burn 1 full and 1 half day for travel...which again for me is very difficult to justify.
We had this very same travel dilema on our upcoming trip and I finally "lost it" and said that's it we are flying...booked 4 roundtrip seats on JetBlue and got a rental car for the 12 days all for just a shade over $1,000. If you think about the wear and tear on the car based on the IRS 34 cents a mile (which is really the only gauge you'd have unless you did a full blown cost of ownership assessment of your own car) almost 3000 miles at .34 is $1,020 cost of wear and tear. So for me it became too good a deal not to fly, even factoring in the schlepping of 2 carseats thru the airport at 6 am...I figure by noon on day one, I'm in the parks...under my driving plan, I'm only halfway to my stopover in HHI.