Can this aircraft be identified by the exposure pattern of its flashing lights?
No, based on the exposure pattern I can't identify the aircraft. But based on the other information you gave, I can. It's an Embraer 175, registration N287SY.



After looking up the exact position of the M13 Globular cluster at the time the photograph was taken, I found that the telescope had to be facing to 64 degrees (East-Northeast), with an elevation angle of 28 degrees. 

I used in-the-sky.org to find the position of the M13 cluster.



I then drew the geodesic from  James River State Park in a heading of 64 degrees
I used academo.org to draw the geodesic on the map



Then using Flight radar 24's history/replay function I found the aircraft


The aircraft is an Embraer 175, flight DL3693,  flying at 38000 feet. 

If we assume:


the telescope is at 500ft MSL, 
pointing up 28 degrees above the horizon, 
the atmosphere is ISA, and
the earth is flat, 


then we can calculate that the aircraft would be approximately 21.5 km horizontally away.

Measuring that out on Google maps gives the following: 



That measurement is perfectly in line with the flight path of the Embraer 175. 

The slant distance to the aircraft is approximately 24.5 kilometres. The track angle of the aircraft is 228 degrees, so we see the aircraft almost from straight ahead.  

The span is 26 meter, I assume the lights are at the wingtips (they are not). The distance between the traces is thus expected to be 3.5 arc minutes.  Close enough!