The tip of the "rocket" (it is actually a scale mockup of a Saturn V that was built in 1999) has an aircraft warning beacon. The beacon has a glass lens protecting it. Typically such protective lenses are cylindrical in shape with a hollow cavity in the middle where the actual beacon light is located. The light from the moon passing through and being refracted by the glass lens appears to be acting as a point source of light. The sun, which was just rising behind the cameras, may also be reflecting off the warning beacon's protective cover.
As some of the comments to the linked video theorize, the light passing through the cylindrical glass lens over the beacon is creating an interference pattern similar to what causes diffraction spikes. The effect is along only one axis due to the cylindrical shape of the glass lens covering the beacon.
From a comment by Ron Jones:
The tip of the rocket acts as a point source and becomes a source for light that has the same coherence of the direct moon light. The interference band is indeed a line of light and dark dots, but they are much dimmer than the moon light and look dark, sort of like the blazingly bright sun spots look dark against a much brighter sun.
The tip of the rocket probably has a nice shiny tip, like the glass dome over a light. At a few miles distance, it becomes essentially a point source.
From a comment by Patrick Bryant:
I think what is happening is the interference of two paths to the eye (or camera), one along the line of sight and another which glances off of a reflective cylinder at the tip of the rocket. The glancing path picks up a 180 degree phase flip but the path length is well within a single wavelength of the normal line of sight path. The result is near total destructive interference for photons coming from the region of the moon perpendicular to the symmetry axis of the cylindrical mirror. Coherence is not needed because even single photons would experience this effect - the two paths interfere with themselves.
From a comment by Paper Burn:
a polarized red airplane warning light on top of the rocket the lens is a disk in a shape causing an optical diffraction Fraunhofer diffraction equation is used to model the diffraction of waves when the diffraction pattern is viewed at a long distance from the diffracting object, and also when it is viewed at the focal plane of an imaging lens.
And Patrick Bryant's reply:
If the cylindrical mirror (shiny metal rod) is vertical, then these two paths exist for the horizontal band perpendicular to the surface of the mirror. The dark band should extend the height of the reflective rod so if you put a taller rod, you could use this effect to make the entire moon dark from that vantage point.
In other comments the original poster of the video (Smarter Every Day) says that the same phenomenon was observed by three different camera/lens combinations: A 300mm lens on a Canon 70D, a 100mm lens on a Panasonic GH5, and a Canon 7D Mark II with unspecified lens that appears, from the video, to be an EF 200-400mm f/4 L IS 1.4X.
The line appears to be slightly tilted with respect to a line perfectly perpendicular to the vertical axis of the rocket mockup. I would not be surprised at all that a close inspection of the glass lens covering the warning beacon would reveal that the glass lens is slightly tilted at the exact same angle.
Regardless of the exact physical phenomenon that explains the line, it seems that the (cylindrical?) glass lens that covers the warning beacon and the moonlight that it refracted/reflected and/or the sunlight that it reflected is the key to what happened.
Any explanation that suggests an atmospheric phenomenon, such as a jet exhaust contrail, ignores the linked video, where the line moves with slight camera movements to remain exactly aligned with the tip of the Saturn V mickup. It also ignores the evidence from three different camera/lens combinations shooting from slightly different positions that all show the phenomenon perfectly aligned with the tip of the rocket.