Problem with high-pass filter - signal gets shifted down
I need the exact real amplitude of each pulse


Then differentiate the signal more thoroughly like this: -



The idea behind this is that you use a "strong" high pass filter that is high enough to decay the DC content within one pulse.

Then measure the peak of the red signal. You can measure it at several points of course and translate each reading to a projected peak reading. You could even low-pass filter in software to recover the shape of the original signal and hence deduce the amplitude.

Try using a simulator also.