Signal distortion in simulation of a current to voltage converter
Diodes such as 1N4148 have a finite impedance even with zero volts across them, the effective resistance can be as low as a few kilohm at close to zero volts. In practice they also will need to be shielded from light that can cause photoelectric effects that can cause issues.
You need diodes that have extremely low leakage - single digit pA.
Other semiconductors such as LEDs or the gates of junction FETs have very low leakage and are sometimes used for this purpose.
The capacitance of the diode will also compromise the stability of the amplifier - the usual way to stabilize the amplifier is, as others have suggested, to put a small capacitor across the feedback resistor.
After all this, it is also probable that the simulation model will not work very well to reflect real-world operation in this area - you may need to physically prototype the circuit.
BTW the gain of a current to voltage converter is expressed in ohms (or V/A). It is not dimensionless.