Thanks Magic. I was running my original circuit with my recently acquired data, and it turns out, like you said, a feedback capacitor wasn't really necessary. Based on the sample data it wasn't running nearly as fast as I thought it would be, so a feedback capacitor isn't necessarily needed. I continued to test my design amongst various amplifiers and diodes, and I've more or less narrowed it down to the following:
OP-AMP: AD795, LT1792, LT1793, AD549, and OPA140.
DIODES: CMOD6001, CMDD6001, BAV116W, BAV199, 1N3595
Running these op-amps with my sample data with each of the various diodes and comparing it to the ideal op-amp model, the difference in output voltages range in the micro-volts. Given the parts limitations and restrictions, I think that's about as best as I can make it.
I was searching around when I found Dan Berard's homemade STM, and it got me thinking about how I need to readjust my board when dealing with pico-amps: using guard loops, voltage bias planes, teflon stand offs, etc. Figuring how to assemble the board appears to be a challenge in itself, moreso than finding components. That low-measurement handbook by Keithley suggested by KeepItSimpleStupid is proving to be useful.
https://dberard.com/home-built-stm/electronics/