Military instructor pilot reported a near miss with a general aviation light aircraft crossing a Military Training Route requiring evasive action.

Narrative

Approx 100 ft pass with early model Aircraft Y while on Low Level Route VR43 between points F and G. Weather was clear; and sun was overhead (non-factor). Aircraft Y was flying SW to NE roughly perpendicular to VR route at approx 1;100 ft AGL.Briefed VR route before flight and visually drew on whiteboard. Leg from Pt. F to Pt. G was specifically mentioned as highest probability of light civil traffic due to prox of multiple uncontrolled airfields. Instructor reminded student 1 min prior Pt. F that next leg would be at 1000' AGL and to 'watch for Aircraft Y'. Front seat pilot at controls was student under instruction. Backseat pilot (myself) was consulting chart on kneeboard and annotating fuel burn (heads down) in the seconds prior to incident. Backseat pilot looked up approx 1 sec prior to pass and spotted Aircraft Y as student simultaneously spotted Aircraft Y and pushed stick forward. Causal factor assessed to be lack of appropriate visual lookout in low altitude/high speed environment. Route was run at 420 kts ground speed.Contributing factors: onboard radar functioned perfectly first 30 min of flight. During low level radar entered unusual failure mode which does not enunciate - simply results in no tracks displayed. Flight was planned in a aircraft model with 2 sticks. That aircraft went down earlier in the day; leading to last minute swap into missioned model with no stick in the back (no controls for backseat pilot). Human factor: Student was looking right and left of jet toward civilian airfields instead of in front of aircraft along it's actual flight path. Emphasis in brief and reminder just prior to that leg of the low level led to student perception of threat coming from left or right instead of actual threat directly in front (which is the backseat's blind spot). Student's inexperience in low altitude environment and inexperience with what to look for likely led to late recognition of Constant-Bearing-Decreasing-Range sight picture associated with mid-air scenario. Combining that with small size of Aircraft Y; 7 mi per minute ground speed; and non-blue-sky-background behind Aircraft Y due to terrain led to near mid-air collision which the backseat pilot was unable to prevent due to lack of controls and lack of time to communicate evasive maneuver.Event led to punitive action within military unit; and safety reporting within military chain. Submitting this report to hopefully raise awareness of VR routes. I'm guessing that Aircraft did not see the thin grey line on his sectional chart; and likely didn't know that high speed aircraft would be at his altitude on that line.As a General Aviation Aircraft owner myself; I hope the other pilot wasn't badly affected.