I spent what is probably too much time this weekend fiddling with the TCP202. I reverse engineered a good portion of the schematic and took a lot of measurements (as a side comment, this board is really weirdly routed, did they use an autorouter or something??). The pulse aberrations are due to the nulling amplifier, nothing in the passive output network of the TCP202 seems to have relevant time constants.
As previous comments have said, the hall effect amplifier is amplified by a OP27 and then further amplified and buffered by a MC1458 + PNP/NPN stage. The schematics for the AM503 and the AM503A have a similar configuration. One thing I noticed was that those units had adjustable feedback on the nulling amplifier output stage and had a R + C feedback network on the hall effect amplifier stage. The TCP202 actually has two sets of unpopulated pads for what I assume is a R + C feedback stage.
I took some interesting waveforms from the TCP202, it is clear that the nulling amplifier is under damped. Here is a capture of the input current (green) probe output (blue) output of the nulling amplifier (orange) and output from the hall effect sensor ( yellow) (orange and yellow should be inverted).
I guess if I took bode plots of a lot of things I could calculate more optimal feedback network values, but I already sunk more time into this than I wanted to and was getting tired of it. So instead I just iterated through possible tweaks of the two feedback networks. I tweaked the feedback resistor on the output stage, lowering it from 10k to 4.3k and that seemed to make the biggest improvement. Installing a 140p + 83k network on the unpopulated pads seemed to contribute a small additional improvement, lots of changes there just made the waveform worse, including RC corner frequencies used in the AM503 and AM503B.
I changed the current level, so the scales do not exactly align, but you can see the waveform is now greatly improved!
I assume manufacturing tolerances on the hall effect sensor lead to a range of gain values, so it seems to make sense that by changing the loop gain elsewhere I could improve the step response. The hall effect resistance in my probe seems to be lower than typical, I after the 220 resistors that drop the voltage from 5V to the hall effect sensor I was only reading ~+-1.8V instead of the +-3V others were mentioning. I think this may be related to why changing the loop gain was beneficial in my case.