From the responses and likely provided functionality, you will almost certainly have to do something "special".
If you shoot in aperture priority mode and add an external 3 or 4 stop neutral density filter between frames you'd achieve your desired effect. While this takes extra mechanical hardware it would be quite easy to achieve for anyone with reasonable mechanical proficiency. Add electronic external release of shutter and filter and a small amount of electronics and a close match to your requirement should be possible.
Several obvious methods come to mind and there will be others.
Dropped "mask" hinged so when released it drops across lens front.
Spring loaded for extra speed.
Released by a number of possible means. If an external shutter release is used then timing of shutter timings and of ND trip could be computer or microcontroller controlled.
Similar but a spring loaded sliding ND filter that slides across lens in a track when released.
Similar but rotating wheel.
Swap in ND filter or change polarisation of polarised filters.
LCD shutter as used for eg auto darken welding goggles.
Whether the cross polariser or LCD goggles are adequate to this task consistently dark I know not, but worth investigating.
How long "immediately after" is will affect what other techniques may work.
Methods that involve changes in camera settings probably require minimum times of a good part of 1 second minimum. Anything sensible would probably require cooperation between dancer and photographer.