I think it would likely be very safe - and probably not as lonely as you might think - there are plenty of other commuters out there, rain and shine and I'd bet some of them have an early day too. Cyclists being mugged or otherwise attacked is a pretty rare event in the first place and I couldn't find any incidents on the Burke specifically.

The only real problem I can recall on a bike path here was a series of attacks on commuters outside of the I-90 tunnel- the muggers were lying in wait at the end of the tunnel and ambushing cycle commuters and stealing their bags. For a while people started waiting for a group to enter and exit the tunnel and eventually the attacks stopped (one intrepid cyclist even rammed the would be thief). I'm not sure if the thieves were eventually caught or not.