Recently I rode home at 3:30 AM from seeing a friend (very good friend I hadn't seen for months, was in town for 12 hours, spent most of those 12 with her!) from just north of the U-district to Queen Anne. It was actually amazingly fun, as I was passed by maybe 3-4 cars the entire trip. Most of the time I was the only one on the road. Closest thing I had to a dangerous encounter was two giant raccoons who glared at me while crossing the street.

By 5AM, that's probably starting to change, but at 4 it's probably similar. Which is to say, unless the Burke is insanely convenient for you, you might find that the roads through the u-district are unusually friendly. I agree with others above that you're likely to be just fine on the BG (weighing the stress the traffic causes you in the car against the possibility of a sketch encounter on the trail, I think you come out ahead on the trail)--but if you're leaving work early enough, roads that would be awful to ride on during the day might be a breeze.