The input current is coming from the tip of an STM. As that current goes through the IVC going through that 10k ohm resistor, Ohm's law, I0 has a voltage. Correct me of I'm wrong, but that's the voltage you're talking about at the input signal source? With regards to that, the max input voltage that we can put at the tip is +-10 volts, so i think we're okay.
Going through some other comments and asking around elsewhere, I kind of realized how dumb I was. Below is a simulation of mine, using the ideal op-amp model and the 1N4184 model (comparing it to the BAV45). Someone mentioned a feedback capacitor, and that seems to magically solve that ringing issue I posted in my previous circuits. Here, I thought it was a bad model, but it was probably the parameters itself. Previously, I didn't add in that feedback capacitor because the original schematic didn't show it, but I guess I need it now if we're switching diodes.
Unfortunately, that brings up the issue of how to get a feedback capacitance that small. Lowest I can get is 100 fF. @magic, I really appreciate all the help you've given me, and I apologize for not realizing this sooner, but in your experience, how would one achieve such a low capacitance? (or maybe it might be easier to use a different diode; then again, this is an ideal op-amp, so perhaps things will change when using a real op-amp model)