I'm not familiar with NIJ IIIA testing specifically, but for ballistic-capture experiments generally you want to use fast-acting (piezoelectric) load sensors to monitor shock waves through all support structures holding your target. That gives you instantaneous-force measurement samples that you can integrate over time to get impulse transmitted through the vest. Do multiple (at least 30, but hundreds would be better) trials with backing and a similar number without. Statistical analysis should tell you how your backing affects impulse transmission.

Energy measurements would be much more difficult because you need to capture various energy-output channels, such as heating, sound and deformation of bullet and vest components. That requires a lot more sensors and uncertainties. It's hardly worth the effort.

I would NOT recommend trying to measure energy absorption because what is important is impulse transmission. The only reason people fixate on energy is that it's more familiar to non-physicists than impulse. To someone on the receiving end of a bullet, impulse is everything and energy is a "don't care." It's impulse (shock wave) that does all the damage.

Thank you so much Charles Masi for your answer and showing me other possible directions.