IMHO, math itself is could be simple as FIR filtering, they mention trapezoidal - quite possible.
The trade between hardware solution (peak detector) and software (fast sampling + FIR integration) is the ratio performance vs cost. I think, that physics of the sensor (XR-100 or whatever) apply certain limitation on the resolution, since detector has an intrinsic capacitance, and RC constant limits fast rising edge of the pulse, making it more "rounded" and consequently hardware ( peak detector) could not precisely measure magnitude. Sharp peak is distorted. Same with first stage of the amplifier, due limitation of the GBW it doesn't make pulse any better, rather worse, smoothing fast edges even more. Though, to get better accuracy the DSP processing seems right way.