Why is my negative peak detector behaving poorly?
There are several points in the AD843 datasheet that appear to be relevant.

I first spotted the "Overdrive Recovery" time (under the general category of "Frequency Response"). Note that the recovery time for positive overdrive (found in your negative peak detector) is significantly longer than the recovery time for negative overdrive (found in your positive peak detector). Half a microsecond when your pulse width is on the order of 2.5 &micro;s (half-cycle @ 200 kHz) could be significant.

Second, the datasheet specifically mentions that the AD843 has trouble driving capacitive loads. Your 10 nF capacitor is more than an order of magnitude larger than any example load they mention in the datasheet.

Third, there is a peak detector circuit given in the datasheet. The topology is slightly different from yours, but more significantly, they use the AD843 in the output stage, but use an AD847 in the input stage "since the AD847 can drive an arbitrarily large value of capacitance".