I've found a better op-amp for I/V conversion - OPA129. This one has similar parameters to the LMC6081, but allows for using +-15V power supply, and swings up to 13V output. This means that for 10Mohm R1 resistor I can swing up to 130nA, and measure as low as 200pA (due to the fact that offset voltage of OPA129 is 2mV).
To lower the offset voltage at the OPA129 output I've used an datasheet-suggested offset correction, that consists of R5, R6, R7, R8 and C9. It allows to trimm the offset using a potentiometer R8. If this will allow to nullify the offset, I guess it should allow me to measure the current and order of magnitude lower, down to ~50pA, or even less - depending on how effective the offset correction will be.
The PCB design is not varying much from the previous one. I've only adjusted several things to match the requirements of the company that is making the PCB. The order was sent to them so in ~2 weeks I should have the board in my hands. I also ordered some elements for the board, including the most important ones: R1 and R3. As R1 I choose 10M 1% Beyschlag resistor with low temperature coefficient - only 50ppm. R3 is 20k 0,1% 15ppm resistor.
As I calculated Vout = Iin*R1*(R4)/(R3), as R4 consists of a resistor and a potentiometer I assume it will be exact 200k. With the said resistors I will get Vout=Iin*10^10, this means 1 volt per 100pA. R1 error is 10^9*1% = 100k and R3 error is 20R. Without temperature effects (when the circuit will be maintained at 25*C) measurement error equals to Iin*1,1*10e6. This is six orders of magnitude lower than the measurement, which is Iin*10e10.
As I want to limit the bandwidth at around 10kHz I'll have to calculate the capacitance of C1 and C8. C1 calculated is 1,6pF and C8 is 79,6pF. Closest values are 1.5pF and 82pF, and such capacitors I bought to use in the final device.
I also ordered both op-amps, so I'll have to wait only for all the parts to come and then I can solder the circuit and do some tests, so stay tuned.